SIAM J. NUMER. ANAL. c 1999 Society for Industrial and Applied Mathematics
Vol. 36, No. 3, pp. 788{818
THE ORTHOGONAL DECOMPOSITION THEOREMS FOR
MIMETIC FINITE DIFFERENCE METHODS
y y
JAMES M. HYMAN AND MIKHAIL SHASHKOV
This paper is dedicated to the fond memory of Ami Harten.
Abstract. Accurate discrete analogs of di erential operators that satisfy the identities and the
orems of vector and tensor calculus provide reliable nite di erence methods for approximating the
solutions to a wide class of partial di erential equations. These methods mimic many fundamental
properties of the underlying physical problem including conservation laws, symmetries in the solution,
and the nondivergence of particular vector elds (i.e., they are divergence free) and should satisfy a
discrete version of the orthogonal decomposition theorem. This theorem plays a fundamental role in
the theory of generalized solutions and in the numerical solution of physical models, including the
Navier{Stokes equations and in electrodynamics. We are deriving mimetic nite di erence approx
imations of the divergence, gradient, and curl that satisfy discrete analogs of the integral identities
satis ed by the di erential operators. We rst de ne the natural discrete divergence, gradient, and
curl operators based on coordinate invariant de nitions, such as Gauss’s theorem, for the divergence.
Next we use the formal adjoints of these natural operators to derive compatible divergence, gradient,
and curl operators with complementary domains and ranges of values. In this paper we prove that
these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate
how a discrete vector can be decomposed into two orthogonal vectors in a unique way, satisfying a
~ ~
discrete analog of the formula A = grad’ + curlB. We also present a numerical example to illus
trate the numerical procedure and calculate the convergence rate of the method for a spiral vector
eld.
Key words. discrete vector analysis, discrete orthogonal decomposition theorem, mimetic nite
di erence methods
AMS subject classi cations. 65N06, 65P05, 53A45
PII. S0036142996314044
1. Introduction. We are developing a discrete analog of vector and tensor cal
culus that can be used to accurately approximate continuum models for a wide range
of physical processes on logically rectangular, nonorthogonal, nonsmooth grids. These
nite di erence methods preserve fundamental properties of the original continuum
di erential operators and allow the discrete approximations of partial di erential equa
tions (PDEs) to mimic critical properties of the underlying physical problem including
conservation laws, symmetries in the solution, and the nondivergence of particular
vector elds (i.e., they are divergence free). These discrete analogs of di erential
operators satisfy the identities and theorems of vector and tensor calculus and are
providing new reliable nite di erence methods for a wide class of PDEs.
The orthogonal decomposition theorem plays a fundamental role in the theory of
generalized solutions and in the numerical approximation of physical models, including
the Navier{Stokes equations and in electrodynamics. In this paper, we will prove
discrete analogs of the orthogonal decomposition theorem for the discretization of a
vector eld. That is, we prove that for given discrete divergence and curl, a discrete
vector can be decomposed into two orthogonal vectors in a unique way satisfying
Received by the editors December 23, 1996; accepted for publication (in revised form) February
18, 1998; published electronically March 30, 1999. This work was performed under the auspices
of the US Department of Energy (DOE) contract W7405ENG36 and the DOE/BES (Bureau of
Energy Sciences) Program in the Applied Mathematical Sciences contract KC070101.
http://www.siam.org/journals/sinum/363/31404.html
y
Los Alamos National Laboratory, T7, MSB284, Los Alamos, NM 87545 (jh@lanl.gov,
misha@t7.lanl.gov).
788DISCRETE ORTHOGONAL DECOMPOSITION 789
~ ~
a discrete analog of the formula A = grad’ + curlB if its normal or tangential
component is given on the boundary.
In [16], we de ned the natural discrete divergence, gradient, and curl operators
based on coordinate invariant de nitions, such as Gauss’s theorem, for the divergence.
We introduced notations for twodimensional (2D) logically rectangular grids, de ned
analogues of line, surface, and volume integrals, described both cell and nodal dis
cretizations for scalar functions, and constructed the natural discretizations of vector
elds, using the vector components normal and tangential to the cell boundaries.
The domains and ranges of the natural discrete operators arise \naturally" from
discrete analogs of Stokes’s theorem. To form secondorder nontrivial combinations
of these operators, which are discrete analogs of div grad, grad div, and curl curl,
the range of the rst operator must be equal to the domain of the second operator.
The natural operators alone are not su cient to construct discrete analogs of the
secondorder operators div grad, grad div, and curl curl because of inconsistencies
in domains and range of values.
In [17], we used the formal adjoints of these natural operators to derive compatible
divergence, gradient, and curl operators with complementary domains and ranges of
values. These new discrete operators are adjoints to the natural operators and, when
combined with natural operators, allow all the compound operators to be constructed.
By construction all of these operators satisfy discrete analogs of the integral identities
satis ed by the di erential operators.
In [16] and [17], we proved discrete analogs of main theorems of vector analysis
for both the natural and adjoint discrete operators. These theorems include Gauss’s
~ ~ ~
theorem; the theorem that divA = 0 if and only if A = curlB; the theorem that
~ ~ ~
curlA = 0 if and only if A = grad’; the theorem that if A = grad’, then the
line integral does not depend on path; and the theorem that if the line integral of a
vector function is equal to zero for any closed path, then this vector is the gradient
of a scalar function.
The discrete analogs of the di erential operators are derived and analyzed using
the supportoperators method (SOM) [33, 35, 36]. In the SOM, rst a discrete ap
proximation is de ned for a rstorder di erential operator, such as the divergence
or gradient, that satis es the appropriate discrete analog of an integral identity, such
as Stokes’s theorem. This initial discrete operator, called the prime operator, then
supports the construction of other discrete operators, using discrete formulations of
the integral identities
Z Z I
~ ~ ~
(1.1) u divWdV + (W; gradu)dV = u (W;~n)dS;
V V @V
Z Z I
~ ~ ~ ~ ~ ~
(1.2) (A; curlB)dV¡ (B; curlA)dV = (~n;A B)dS:
V V @V
For example, if the initial discretization is de ned for the divergence ( prime operator),
it should satisfy a discrete form of Gauss’s theorem. This prime discrete divergence,
DIV; is then used to support the derived discrete operator GRAD satisfying a discrete
version of the integral identity (1.1). The derived operator GRAD would then also
be the negative adjoint of DIV.
The SOM uses the discrete versions of integral identities as a basis to construct
discrete operators with compatible domains and ranges and is used to extend the
set of discrete operators by forcing discrete analogs of the integral identities (1.1) or
(1.2). In the simplest case, when boundary integrals vanish, the above entities imply790 J. HYMAN AND M. SHASHKOV
div =¡grad and curl = curl in the sense of the inner products
Z Z
~ ~ ~ ~
(1.3) (u;v) = uvdV ; (A;B) = (A;B)dV :
H H
V V
The natural discrete divergence is de ned as a discrete operator from the space HS
of discrete vector functions, de ned by their orthogonal projections onto directions
perpendicular to the face of the cell, to the space HC of discrete scalar functions,
given by their values in the cell (see [16] and section 3 of this paper for details)
DIV :HS!HC:
The discrete gradient,
GRAD :HC!HS;
is constructed as a negative adjoint to DIV,
(1.4) GRAD =¡DIV ;
where we have indicated the derived adjoint operator by the overbar.
The natural discrete gradient GRAD is de ned as a discrete operator from space
HN of discrete scalar functions, given by their values in the nodes, to the spaceHL
of discrete vector functions, de ned by their orthogonal projections onto directions of
the edges of the cell
GRAD :HN!HL:
The discrete divergence,
DIV :HL!HN;
is constructed as a negative adjoint to GRAD,
(1.5) DIV =¡GRAD :
In a similar way, the natural CURL :HL!HS is used to construct another discrete
curl,
(1.6) CURL :HS!HL; CURL = CURL :
These operators satisfy the main discrete theorems of vector analysis [16, 17].
The natural operators alone can be combined only to construct the trivial oper
ators:
(1.7) DIV CURL :HL!HC; DIV CURL 0;
(1.8) CURL GRAD :HN!HS; CURL GRAD 0:
For example, we cannot apply DIV to GRAD because the range of values of GRAD
does not coincide with the domain of operator DIV, and so on.
Similarly, the adjoint operators alone can be combined only to construct the trivial
operators:
(1.9) DIV CURL :HS!HN; DIV CURL 0;
(1.10) CURL GRAD :HC!HL; CURL GRAD 0:DISCRETE ORTHOGONAL DECOMPOSITION 791
However, the natural and adjoint operators together can be combined to form all the
nontrivial highorder operators:
(1.11) DIV GRAD :HC!HC; DIV GRAD :HN!HN;
(1.12) CURL CURL :HS!HS; CURL CURL :HL!HL;
(1.13) GRAD DIV :HL!HL; GRAD DIV :HS!HS:
In this paper, we use results from [16, 17] to prove discrete analogs of one of
the most important theorems of vector analysis, i.e., the orthogonal decomposition
theorem for vector elds. This theorem states that a vector function ~a in a bounded
domain, V , is uniquely determined by its divergence, curl, and values of normal
component on the boundary:
(1.14) div~a = ; (x;y;z)2V;
(1.15) curl~a =!; ~ (x;y;z)2V;
and
(1.16) (~a;~n)= ; (x;y;z)2@V :
Moreover, any vector~a can be represented in the form
~
(1.17) ~a = grad’ + curlB:
This decomposition forms a considerable part of the theory of generalized so
lutions [41], and Weyl’s theorem on orthogonal decomposition plays a fundamental
role in solving the Navier{Stokes equations [12, 19, 21, 40, 41] and in electrodynamics
[15]. The discrete version of Weyl’s theorem, formulated in [39] for square grids in two
dimensions, can be used to construct highquality nitedi erence methods (FDMs)
for Navier{Stokes equations [3, 4]. The orthogonal decomposition theorem has been
proved for discrete vector elds on both orthogonal grids [8, 22, 23, 24, 20, 32] and
Voronoi grids [27, 28, 14]. Similar results for the niteelement method can be found
in [12, 26]. The orthogonal decomposition theorem has been applied in numerical
simulation to remove arti cial vorticity from a velocity vector eld [10]. Some general
results for discrete models can also be found in [6, 7].
The discrete analog of the orthogonal decomposition theorem for vectors from the
~
spaceHS grids states that any discrete vector function A2HS, where
h h
~ ~
(1.18) DIVA = ; CURLA =! ;
~ ~
where the normal component of the vector A is given on the boundary (for A2HS
these are the components used to describe discrete vector eld) and can be represented
in the form
~ ~
(1.19) A = GRAD’ + CURLB;
~
where’2HC andB2HL. As in the continuous case, the discrete vector functions
~
GRAD’ and CURLB are orthogonal to each other.
~
The theorem also holds for any discrete vector function A2HL, where
h h
~ ~
(1.20) DIVA = ; CURLA =! :792 J. HYMAN AND M. SHASHKOV
(1,N)
(1,N)
(M,N)
(M,N)
i,N
i,N
U
U i,N
i+1/2,N
j
j
U
U
i+1/2,j+1/2 U M,j
M,j+1/2
U
M,j M,j
i,j
i,j
U i,j
1,j
1,j
1,j+1/2
U
1,j
(M,1)
(M,1)
i,1
i,1
U U
i+1/2,1 (1,1) i,1
(1,1)
i
i
(a) Cellcentered grid,HC (b) Nodal grid,HN
Fig. 2.1. On a logically rectangular grid, the scalar function values can be either cellcentered
(HC), as in (a), or de ned at the nodes ( HN), as in (b).
~ ~
Here the tangential component of the vectorA is given on the boundary (forA2HL
these are the components used to describe discrete vector eld) and can be presented
in the form
~ ~
(1.21) A = GRAD’ + CURLB;
~ ~
where’2HN andB2HS; and the discrete vector functions GRAD’ and CURLB
are orthogonal to each other.
The discrete orthogonal decomposition theorem can be viewed from two di erent
perspectives. In the rst case, we can consider the solution of the discrete problem
(1.18) as an approximation to the solution of the corresponding continuous problem
(1.14), (1.15), (1.16). In the second case, we have a pure discrete problem: for a given
~
discrete vector, A2HS, nd its representation in the form (1.19), or for the vector
~
A2HL, nd its representation in the form (1.21).
After describing the grid, discretizations of scalar and vector functions, and inner
products in spaces of discrete functions, we will review the derivation of the natural
and adjoint nite di erence analogs for the divergence, gradient, and curl. Next we
prove the discrete orthogonal decomposition theorems for discrete vector functions
fromHS andHL. Finally, we present a numerical example for the orthogonal de
composition of a spiral vector eld and demonstrate a secondorder convergence rate
~
in max norm for the recovered (reconstructed) discrete vector function A2HS.
2. Spaces of discrete functions.
2.1. Grid. We index the nodes of a logically rectangular grid using (i;j), where
1 i M and 1 j N (see Figure 2.1). The quadrilateral de ned by the nodes
(i;j), (i+1;j), (i+1;j + 1), and (i;j + 1) is called the (i+1=2;j +1=2) cell (see
Figure 2.2 (a)). The area of the (i+1=2;j+1=2) cell is denoted byVC , the
i+1=2;j+1=2
length of the side that connects the vertices (i;j) and (i;j +1) is denoted byS ,
i;j+1=2
and the length of the side that connects the vertices (i;j) and (i+1;j) is denoted by
S . The angle between any two adjacent sides of cell (i;j) that meet at node
i+1=2;j
i+1=2;j+1=2
(k;l) is denoted by ’ .
k;lDISCRETE ORTHOGONAL DECOMPOSITION 793
z i,j+1,k+1
i+1,j+1,k+1
i,j,k+1
( i+1,j+1 )
i+1,j,k+1
( i,j+1 )
y
i,j+1,k
VC
x
S
i+1/2,j+1/2
i,j+1/2
i+1/2,j+1/2
j i,j,k
i+1,j+1,k
i+1,j
Sh
i+1,j,k
( i+1,j )
i+1/2,j
( i,j ) x
(a) 2D grid (b) 3D grid
Fig. 2.2. (a) The (i+1=2;j +1=2) cell in a logically rectangular grid has areaVC
i+1=2;j+1=2
and sidesS ,S ,S , andS . The interior angle betweenS
i;j+1=2 i+1=2;j i+1;j+1=2 i+1=2;j+1 i+1=2;j
i+1=2;j+1=2
and S is ’ . (b) The 2D (i+1=2;j +1=2) cell (z =0) is interpreted as the
i+1;j+1=2
i+1;j
base of a 3D logically cuboid (i+1=2;j +1=2;k+1=2) cell (a prism) with unit height.
i,j,k+1
z
i,j+1,k
h
i,j,k
x
i+1,j,k
Fig. 2.3. The ( ; ; ) curvilinear coordinate system is approximated by thei;j, andk piecewise
linear grid lines.
When de ning discrete di erential operators, such as CURL, it is convenient to
consider a 2D grid as the projection of a threedimensional (3D) grid. This approach
eliminates any ambiguity in the notation and simpli es generalizing the FDMs to 3D.
In this paper, we consider functions of the coordinates x and y and extend the grid
into a third dimension, z, by extending a grid line of unit length into the z direction
to form a prism with unit height and with a 2D quadrilateral cell as its base (see
Figure 2.2 (b)).
Sometimes it is useful to interpret the grid as being formed by intersections of
broken lines that approximate the coordinate curves of some underlying curvilinear
coordinate system ( , , ). The , ,or coordinate corresponds to the grid line
where the index i, j,or k is changing, respectively (see Figure 2.3).
We denote the length of the edge (i;j;k)¡ (i+1;j;k)by l , the length
i+1=2;j;k
of the edge (i;j;k)¡ (i;j +1;k)by l , and the length of the edge (i;j;k)¡
i;j+1=2;k
(i;j;k +1) by l (which we have chosen to be equal to 1). The area of the
i;j;k+1=2
surface (i;j;k)¡(i;j+1;k)¡(i;j;k+1)¡(i;j+1;k+1), denoted byS ,is
i;j+1=2;k+1=2
the analog of the element of the coordinate surfacedS . Similarly, the area of surface
(i;j;k)¡ (i+1;j;k)¡ (i;j;k+1)¡ (i+1;j;k + 1) is denoted byS .We
i+1=2;j;k+1=2
use the notation S for the area of the 2D cell (i+1=2;j +1=2); that
i+1=2;j+1=2;k
is, S =VC . Because the arti cially constructed 3D cell is a
i+1=2;j+1=2;k i+1=2;j+1=2794 J. HYMAN AND M. SHASHKOV
right prism with unit height, we have
S =l l =l
i;j+1=2;k+1=2 i;j+1=2;k i;j;k+1=2 i;j+1=2;k
and
S =l l =l :
i+1=2;j;k+1=2 i+1=2;j;k i;j;k+1=2 i+1=2;j;k
With this 3D interpretation, the 2D notations S and S are not
i;j+1=2 i+1=2;j
ambiguous because the 3D surface (i;j;k); (i;j +1;k); (i;j;k +1); (i;j +1;k +1)
corresponds to an element of the coordinate surfaceS , and since the prism has unit
height, the length of the side (i;j)¡ (i;j + 1) is equal to the area of the element of
this coordinate surface.
2.2. Discrete scalar and vector functions. In a cellcentered discretization,
the discrete scalar function U is de ned in the space HC and is given
i+1=2;j+1=2
by its values in the cells (see Figure 2.1 (a)), except at the boundary cells. The
treatment of the boundary conditions requires introducing scalar function values at
the centers of the boundary segments: U ;U , wherej =1;:::;N¡ 1,
(1;j+1=2) (M;j+1=2)
and U ;U , where i =1;:::;M¡ 1 . In three dimensions the cell
(i+1=2;1) (i+1=2;N)
centered scalar functions are de ned in the centers of the 3D prisms, except in the
boundary cells, where they are de ned on the boundary faces. The 2D case can
be considered a projection of these values onto the 2D cells and midpoints of the
boundary segments.
In a nodal discretization, the discrete scalar function U is de ned in the space
i;j
HN and is given by its values in the nodes (see Figure 2.1 (b)). The indices vary in
the same range as for coordinates x ;y .
i;j i;j
We assume that vectors may have three components, but in our 2D analysis,
the components depend on only two spatial coordinates, x and y. We consider two
di erent spaces of discrete vector functions for our 3D coordinate system. The HS
space (see Figure 2.4 (a)), where the vector components are de ned perpendicular
to the cell faces, is the natural space when the approximations are based on Gauss’
divergence theorem. TheHL space (see Figure 2.5 (a)), where the vectors are de ned
tangential to the cell edges, is natural for approximations based on Stokes’ circulation
theorem.
The projection of the 3DHS vector discretization space into two dimensions
results in the face vectors being de ned perpendicular to the quadrilateral cell sides
and in the cellcentered vertical vectors being de ned perpendicular to the 2D plane
(see Figure 2.4 (b)). We use the notation
WS :i=1;:::;M ; j =1;:::;N¡ 1
(i;j+1=2)
for the vector component at the center of faceS (sidel ), the notation
(i;j+1=2) (i;j+1=2)
WS :i=1;:::;M¡1; j =1;:::;N
(i+1=2;j)
for the vector component at the center of face S (side l ), and the
(i+1=2;j) (i+1=2;j)
notation
WS :i=1;:::;M¡1; j =1;:::;N¡ 1
(i+1=2;j+1=2)
for the component at the center of faceS (2D cell V ).
(i+1=2;j+1=2) i+1=2;j+1=2DISCRETE ORTHOGONAL DECOMPOSITION 795
i,j+1,k+1
z
i+1,j+1,k+1
i,j,k+1
( i+1,j+1 )
h
WS
i+1/2,j+1
i+1,j,k+1
x
WS
i,j+1/2,k+1/2
( i,j+1 )
i,j+1,k
y
WS
x
i+1,j+1/2
h
WS
WS
i+1/2,j,k+1/2
x
i,j+1/2
WSz
i+1/2,j+1/2
i,j,k
WSz i+1,j+1,k
i+1/2,j+1/2,k
i+1,j,k
h
WS ( i+1,j )
i+1/2,j
x
( i,j )
(a)HS in 3D (b)HS in 2D
Fig. 2.4. (a)HS discretization of a vector in three dimensions. (b) 2D interpretation of the
HS discretization of a vector results in the face vectors being de ned perpendicular to the cell sides
and the vertical vectors being de ned at cell centers perpendicular to the plane.
i,j+1,k+1
z
i+1,j+1,k+1
i,j,k+1
( i+1,j+1 )
i+1,j,k+1
( i,j+1 )
y
z
WL
i,j,k+1/2
i,j+1,k
W h
L
i,j+1/2
h
WL
i,j+1/2,k
i,j,k
i+1,j+1,k
WLx
i+1/2,j,k
WLz
i,j
i+1,j,k
( i+1,j )
WLx
x ( i,j )
i+1/2,j
(a)HL in 3D (b)HL in 2D
Fig. 2.5. (a)HL discretization of a vector in three dimensions. (b) 2D interpretation of the
HL discretization of a vector results in the edge vectors tangential to the cell sides and the vertical
vectors being de ned at cell nodes.
The projection of the 3DHL vector discretization space into two dimensions
results in the vectors being de ned as tangential to the quadrilateral cell sides and in
a vertical vector at the nodes (see Figure 2.5 (b)). We use the notation
WL :i=1;:::;M¡1; j =1;:::;N
(i+1=2;j)
for the component at the center of edge l (in 2D, the same position as for
(i+1=2;j)
WS , the notation
(i+1=2;j))
WL :i=1;:::;M ; j =1;:::;N¡ 1
(i;j+1=2)
for the component at the center of edge l (in 2D, the same position as for
(i;j+1=2)
WS ), and the notation
(i;j+1=2)796 J. HYMAN AND M. SHASHKOV
WL :i=1;:::;M ; j =1;:::;N
(i;j)
for the component at the center of edge l (in 2D the position that corresponds
(i;j)
to node (i;j)).
From here on, there will not be any dependence on thek index, and it is dropped
from the notation.
2.3. Discrete inner products. In the space of discrete scalar functions, HC,
(functions de ned in the cell centers), the natural inner product corresponding to the
continuous inner product
Z I
(2.1) (u;v) = uvdV + uvdV
H
V @V
is
M¡1 N¡1
X X
(U;V ) = U V VC
HC (i+1=2;j+1=2) (i+1=2;j+1=2) (i+1=2;j+1=2)
i=1 j=1
M¡1 N¡1
X X
(2:2) + U V S + U V S
(i+1=2;1) (i+1=2;1) (i+1=2;1) (M;j+1=2) (M;j+1=2) (M;j+1=2)
i=1 j=1
M¡1 N¡1
X X
+ U V S U V S :
(i+1=2;N) (i+1=2;N) (i+1=2;N) (1;j+1=2) (1;j+1=2) (1;j+1=2)
i=1 j=1
0
In our future consideration, we will use the subspace ofHN,HN, where discrete
functions are equal to zero on the boundary:
0
def
HN = fU2HN;U = 0 on the boundaryg
i;j
(the notation of \zero" above the name of a space indicates the subspace where the
functions are equal to zero on the boundary), with the inner product de ned as
M¡1 N¡1
X X
def
(2.3) (U;V ) = U V VN ;
0
(i;j) (i;j) (i;j)
HN
i=2 j=2
whereVN is the nodal volume.
(i;j)
In the space of vector functionsHS, the natural inner product corresponding to
the continuous inner product (3.14) is
M¡1N¡1
X X
~ ~ ~ ~
(2.4) (A;B) = (A;B) VC ;
HS (i+1=2;j+1=2) (i+1=2;j+1=2)
i=1 j=1
~ ~
where (A;B) is the dot product of two vectors. Next, we de ne this dot product in
terms of the components of the vectors perpendicular to the cell sides (see Figure
2.6). Suppose that the axes and form a nonorthogonal basis and that ’ is the
~ ~
angle between these axes. If the unit normals to the axes are nS and nS , then
~
the components of the vector W in this basis are the orthogonal projections WS DISCRETE ORTHOGONAL DECOMPOSITION 797
h
nS
h lh
WLh
W
WSh
x
j l
x
WL
x
WSx
nSx
Fig. 2.6. The grid lines ( ; ) form a local nonorthogonal coordinate system with unit vectors
~ ~ ~
~
l ; l and corresponding unit normals to these directions,nS andnS . In this basis, the components
~
(WL ;WL ) of vector W are the orthogonal projections onto the grid lines, and the components
(WS ;WS ) are the orthogonal projections to the normal directions.
~
and WS of W onto the normal vectors. The expression for the dot product of
~ ~
A=(AS ;AS ) and B =(BS ;BS )is
AS BS +AS BS +(AS BS +AS BS ) cos’
~ ~
(2.5) (A;B)= ;
2
sin ’
where ’ is the angle between these axes (see Figure 2.6).
From this expression, the dot product in the cell is approximated by
(i+1=2;j+1=2)
1
X V
(i+k;j+l)
~ ~
(A;B) =
(i+1=2;j+1=2)
(i+1=2;j+1=2)
2
sin ’
k;l=0
(i+k;j+l)
h
(2.6) AS BS +AS BS
(i+k;j+1=2) (i+k;j+1=2) (i+1=2;j+l) (i+1=2;j+l)
k+l
+(¡1) (AS BS
(i+k;j+1=2) (i+1=2;j+l)
i
(i+1=2;j+1=2)
+AS BS )cos’ ;
(i+1=2;j+l) (i+k;j+1=2)
(i+k;j+l)
(i+1=2;j+1=2)
where the weights V satisfy
(i+k;j+l)
1
X
(i+1=2;j+1=2) (i+1=2;j+1=2)
(2.7) V 0; V =1:
(i+k;j+l) (i+k;j+l)
k;l=0
In this formula, each index (k;l) corresponds to one of the vertices of the (i+1=2;j +
1=2) cell, and notations for weights are the same as for angles of cell.
The inner product inHL is similar to the inner product for spaceHS:
M¡1N¡1
X X
~ ~ ~ ~
(2.8) (A;B) = (A;B) VC ;
HL
(i+1=2;j+1=2) (i+1=2;j+1=2)
i=1 j=1
~ ~
where (A;B) approximates the dot product of two vectors at the cell (i +
i+1=2;j+1=2
1=2;j +1=2). InHL, the vectors are represented by orthogonal projections to the798 J. HYMAN AND M. SHASHKOV
directions of the edges of the 3D cells (see Figure 2.6). If the axes and form a
~
nonorthogonal basis, the components of the vectorW in this basis are the orthogonal
~ ~
projectionsWL andWL of W onto the directions of the coordinate axes. If A =
~
(AL ;AL ) and B =(BL ;BL ), then the dot product is
AL BL +AL BL ¡ (AL BL +AL BL ) cos’
~ ~
(2.9) (A;B)= :
2
sin ’
From a formal point of view, the only di erence between this formula and the one
for the surface components (see (2.5)) is the minus sign before the third term. This
di erence can be easily understood by taking into account that the basis vectors of
the nonorthogonal local systems are perpendicular to each other.
Equation (2.9) is used to approximate the dot product in a cell:
(i+1=2;j+1=2)
1
X
V
(i+k;j+l)
~ ~
(A;B) =
(i+1=2;j+1=2)
(i+1=2;j+1=2)
2
sin ’
k;l=0
(i+k;j+l)
h
(2.10) AL BL +AL BL
(i+1=2;j+l) (i+1=2;j+l) (i+k;j+1=2) (i+k;j+1=2)
¡
k+l
¡(¡1) AL BL
(i+1=2;j+l) (i+k;j+1=2)
i
(i+1=2;j+1=2)
+AL BL cos’ ;
(i+k;j+1=2) (i+1=2;j+l)
(i+k;j+l)
(i+1=2;j+1=2)
where V represents the same weights used for the spaceHS.
(i+k;j+l)
When computing the adjoint relationships between the discrete operators, it is
helpful to introduce the formal inner products (denoted by square brackets [ ; ]) in
the spaces of scalar and vector functions. InHC, the formal inner product is
M¡1 N¡1 M¡1
X X X
[U;V ] = U V + U V
HC (i+1=2;j+1=2) (i+1=2;j+1=2) (i+1=2;1) (i+1=2;1)
i=1 j=1 i=1
N¡1 M¡1 N¡1
X X X
+ U V + U V + U V ;
(M;j+1=2) (M;j+1=2) (i+1=2;N) (i+1=2;N) (1;j+1=2) (1;j+1=2)
j=1 i=1 j=1
inHS, the formal inner product is
M N¡1 M¡1 N
XX XX
~ ~
[A;B] = AS BS + AS BS :
HS
(i;j+1=2) (i;j+1=2) (i+1=2;j) (i+1=2;j)
i=1 j=1 i=1 j=1
The natural and formal inner products satisfy the relationships
~ ~ ~ ~
(2.11) (U;V ) =[CU; V ] and (A;B) =[SA; B] ;
HC HC HS HS
whereC andS are symmetric positive operators in the formal inner products. For
operatorC we have
(2.12) [CU;V ] =[U;CV ] and [CU;U] > 0;
HC HC HC
and therefore
(CU) = VC U ;
(i+1=2;j+1=2) (i+1=2;j+1=2) (i+1=2;j+1=2)
i=1;:::;M¡ 1;j =1;:::;N¡ 1;
(CU) = S U ;i = 1 and i =M; j =1;:::;N¡ 1;
(i;j+1=2) (i;j+1=2) (i;j+1=2)
(CU) = S U ;i=1;:::;M¡ 1;j = 1 andj =N:
(i+1=2;j) (i+1=2;j) (i+1=2;j)DISCRETE ORTHOGONAL DECOMPOSITION 799
i,j+1
i+1,j
i,j
i,j
Stencil for operator S
Stencil for operator S
21
12
Fig. 2.7. The stencils of the componentsS andS of the symmetric positive operatorS that
12 21
~ ~ ~ ~
connects the natural and formal inner products (A;B) =[SA;B] .
HS HS
The operatorS can be written in block form,
S S AS S AS +S AS
11 12 11 12
~
(2.13) SA = = ;
S S AS S AS +S AS
21 22 21 22
and is symmetric and positive in the formal inner product
~ ~ ~ ~ ~ ~
(2.14) [SA;B] =[A;SB] ; [SA;A] > 0:
HS HS HS
By comparing the formal and natural inner products
M N¡1
XX
~ ~ ~ ~
(A;B) =[SA;B] = [(S AS ) +(S AS ) ]BS
HS HS 11 (i;j+1=2) 12 (i;j+1=2) (i;j+1=2)
i=1 j=1
M¡1 N
XX
(2:15) + [(S AS ) +(S AS ) ]BS
21 22
(i+1=2;j) (i+1=2;j) (i+1=2;j);
i=1 j=1
we can derive the explicit formulas forS:
0 1
(i+k;j+1=2)
X
V
(i;j+l)
@ A
(S AS ) = AS ;
11 (i+k;j+1=2) (i;j+1=2)
(i;j+1=2)
2
sin ’
(i;j+l)
k= 1=2;l=0;1
(S AS )
12
(i;j+1=2)
(i+k;j+1=2)
X
V
1 (i+k;j+1=2)
(i;j+l)
k+ +l
2
= (¡1) cos’ AS ;
(i+k;j+1=2) (i+k;j+l)
2 (i;j+l)
sin ’
(i;j+l)
k= 1=2;l=0;1
(2.16)
(S AS )
21
(i+1=2;j)
(i+1=2;j+k)
X
V
1
(i+1=2;j+k)
k+ +l (i+l;j)
2
= (¡1) cos’ AS ;
(i+1=2;j+k) (i+l;j+k)
2 (i+l;j)
sin ’
(i+l;j)
k= 1=2;l=0;1
0 1
(i+1=2;j+k)
X
V
(i+l;j)
@ A
(S AS ) = AS :
22 (i+1=2;j+k) (i+1=2;j)
(i+1=2;j)
2
sin ’
(i+l;j)
k= 1=2;l=0;1
The operatorsS andS are diagonal, and the stencils for the operatorsS and
11 22 12
S are shown on Figure 2.7. These formulas are valid only for sides of the grid cells
21
interior to the domain. They can be applied at the sides on the domain boundary
if the grid and discrete functions are rst extended to a row of points outside the
domain by using the appropriate boundary conditions.800 J. HYMAN AND M. SHASHKOV
0
The relationship between the natural and formal inner products inHN is
(2.17) (U;V ) =[NU; V ] ;
0 0
HN HN
whereN is the symmetric positive operator in the formal inner product,
(2.18) [NU;V ] =[U;NV ] ; [NU;U] > 0;
HN HN HN
and
(2.19) (NU) =VN U ;i=2;:::;M¡ 1;j =2;:::;N¡ 1:
(i;j) (i;j) (i;j)
The operatorL, which connects the formal and natural inner products inHL
(similar to operatorS for spaceHS), can be written in block form as
L L AL L AL +L AL
11 12 11 12
~
(2.20) LA = = :
L L AL L AL +L AL
21 22 21 22
This operator is symmetric and positive in the formal inner product:
~ ~ ~ ~ ~ ~
(2.21) [LA;B] =[A;LB] ; [LA;A] > 0:
HL HL HL
A comparison of formal and natural inner products gives the following:
0 1
(i+1=2;j+k)
X
V
(i+l;j)
@ A
(L AL ) = AL ;
11
(i+1=2;j)
(i+1=2;j)
(i+1=2;j+k)
2
sin ’
1
k= ;l=0;1
(i;j+l)
2
(L AL )
12
(i+1=2;j)
(i+1=2;j+k)
X V
1
(i+l;j) (i+1=2;j+k)
k+ +l
2
=¡ (¡1) cos’ AL ;
(i+l;j+k)
(i+1=2;j+k) (i+l;j)
2
sin ’
1
k= ;l=0;1 (i+l;j)
2
(2.22)
(L AL )
21
(i;j+1=2)
(i+k;j+1=2)
X
V
1 (i;j+l)
(i+k;j+1=2)
k+ +l
2
=¡ (¡1) cos’ AL ;
(i+k;j+l)
(i;j+l)
(i+k;j+1=2)
2
sin ’
1
k= ;l=0;1
(i;j+l)
2
0 1
(i+k;j+1=2)
X V
(i;j+l)
@ A
(L AL ) = AL :
22 (i;j+1=2)
(i;j+1=2)
(i+k;j+1=2)
2
sin ’
1
k= ;l=0;1
(i;j+l)
2
The operators L and L are diagonal, and the stencils for the operators L and
11 22 21
L (in 2D) are the same as for the operators S and S (see Figure 2.7).
12 12 21
These discrete inner products satisfy the axioms of inner products; that is,
(A; B) =(B;A) ;
H H
h h
( A;B ) = (A; B) (for all real numbers );
H H
h h
(A +A;B) =(A B) +(A B) ;
1 2 H 1 H 2 H
h h h
(A; A) 0 and (A; A) = 0 if and only if A=0:
H H
h h
In these axioms, A and B are either discrete scalar or discrete vector functions and
( ; ) is the appropriate discrete inner product.
H
h
Therefore the discrete inner products are true inner products and not just ap
proximations of the continuous inner products. Also, discrete spaces are Euclidean
spaces.DISCRETE ORTHOGONAL DECOMPOSITION 801
3. Discrete analogs of div; grad, and curl.
3.1. Natural operators.
3.1.1. Natural operator DIV. The coordinate invariant de nition of the div
operator is based on Gauss’s divergence theorem:
H
~
(W;~n)dS
@V
~
(3.1) divW = lim ;
V!0
V
where ~n is a unit outward normal to boundary @V . To extend the operator div to
the boundary, we de ne the extended divergence operator d as
+divw; ~ (x;y)2V;
(3.2) dw~ =
¡(w; ~ ~n); (x;y)2@V:
The corresponding natural de nition of the discrete divergence operator is
(3.3) DIV :HS!HC;
where
~
(3.4) (DIVW )
(i+1=2;j+1=2)
¡
1
= WS S ¡WS S
(i+1;j+1=2) (i+1;j+1=2) (i;j+1=2) (i;j+1=2)
VC
(i;j)
¡ “
+ WS S ¡WS S :
(i+1=2;j+1) (i+1=2;j+1) (i+1=2;j) (i+1=2;j)
The discrete operator DIV is also extended to the boundary, and we denote the
~
extended operator as D. This operator coincides with DIV, (DW ) =
(i+1=2;j+1=2)
~
(DIVW ) on the internal cells and is de ned on the boundary by
(i+1=2;j+1=2)
~
(DW ) =¡WS ;i=1;:::;M¡ 1;
(i+1=2;1) (i+1=2;1)
~
(3.5) (DW ) =+WS ;i=1;:::;M¡ 1;
(i+1=2;N) (i+1=2;N)
~
(DW ) =¡WS ;j =1;:::;N¡ 1;
(1;j+1=2) (1;j+1=2)
~
(DW ) =+WS ;j =1;:::;N¡ 1:
(M;j+1=2) (M;j+1=2)
3.1.2. Natural operator GRAD. For any directionl given by the unit vector
~
l, the directional derivative can be de ned as
@u
~
(3.6) =(gradu;l):
@l
For a functionU 2HN, this relationship leads to the coordinate invariant de nition
i;j
of the natural discrete gradient operator:
(3.7) GRAD :HN!HL:
~
The vector G = GRADU is de ned as
U ¡U U ¡U
i+1;j i;j i;j+1 i;j
(3.8) GL = ; GL = ; GL =0:
i;j
i+1=2;j i;j+1=2
l l
i+1=2;j i;j+1=2
Note that if GRADU = 0, then scalar discrete functionU is a constant, and vice
versa.802 J. HYMAN AND M. SHASHKOV
3.1.3. Natural operator CURL. The coordinate invariant de nition of the
curl operator is based on Stokes’s circulation theorem,
H
~ ~
(B;l)dl
l
~
(3.9) (~n; curlB) = lim ;
S!0
S
where S is the surface spanning (based on) the closed curve l, ~n is the unit outward
~
normal to S, and l is the unit tangential vector to the curve l.
Using the discrete analog of (3.9), we can obtain expressions for components of
~ ~
vector R =(RS ;RS ;RS ) = CURLB; where
CURL :HL!HS;
and
BL l ¡BL l
i;j+1 i;j+1;k+1=2 i;j i;j;k+1=2
(3.10) RS =
i;j+1=2
S
i;j+1=2;k+1=2
BL ¡BL
i;j+1 i;j
= ;
l
i;j+1=2
BL l ¡BL l
i+1;j;k+1=2 i+1;j;k+1=2 i;j;k+1=2 i;j;k+1=2
(3.11) RS =¡
i+1=2;j
S
i+1=2;j;k+1=2
BL ¡BL
i+1;j i;j
=¡ ;
l
i+1=2;j
and
RS
i+1=2;j+1=2
¡
= BL l ¡BL l ¡
i+1;j+1=2 i+1;j+1=2 i;j+1=2 i;j+1=2
¡ “
BL l ¡BL l =S
i+1=2;j+1 i+1=2;j+1 i+1=2;j i+1=2;j i+1=2;j+1=2
(3.12)
(see [16] for details).
~ ~
Note that if B =(0; 0;BL ), then the condition CURLB = 0 is equivalent to
~
BL = const. This follows since for any such vector B we have that RS =0,
i;j i;j
and from (3.10), (3.11) we can conclude that RS and RS equal zero only if
i;j i;j
BL = const.
i;j
3.1.4. Properties of natural discrete operators. The natural discrete oper
ators satisfy discrete analogs of the theorems of vector analysis [16], including Gauss’s
~ ~ ~
theorem; the theorem that divA = 0 if and only if A = curlB; the theorem that
~ ~ ~
curlA = 0 if and only if A = grad’; the theorem that if A = grad’, then the
line integral does not depend on path; and the theorem that if the line integral of a
vector function is equal to zero for any closed path, then this vector is the gradient
of a scalar function.
In this paper we use two theorems:
~ ~ ~ ~ ~
DIVA = 0 if and only if A = CURLB, where A2HS and B2HL;
~ ~ ~
CURLA = 0 if and only if A = GRADU, where A2HL and U2HN.DISCRETE ORTHOGONAL DECOMPOSITION 803
3.2. Adjoint operators. In [17], we use the SOM to derive the operators DIV;
GRAD, and CURL from discrete analogs of the integral identities (1.1) and (1.2).
These identities connect the di erential operators div, grad, and curl and allow us
to obtain their discrete analogs with consistent domains and ranges of values.
3.2.1. Operator GRAD. To construct the operator GRAD we use a discrete
analog of the identity (1.1). If we introduce the space of scalar functions H with the
inner product
Z I
(3.13) (u; v) = uvdV + uvdS; u;v2H;
H
V @V
and the space of vector functions H with inner product de ned as
Z
~ ~ ~ ~ ~ ~
(3.14) = (A; B)dV ; A;B2 H;
(A; B)
H
V
then the identity (1.1) implies
(3.15) d =¡grad
or
Z I
(dw; ~ u) = u divwd ~ V¡ u (w; ~ ~n)dS
H
V @V
Z
(3.16) =¡ (w; ~ gradu)dV
V
=(w; ~ ¡gradu) :
H
We will base the de nition of GRAD on a discrete analog of (3.14). In section
3.1.1, we de ned the operator D as the discrete analog of the continuous operator
def
d. Therefore the derived gradient operator is de ned by GRAD = ¡D , where the
def
adjoint is taken in the natural inner products (from here on, we will use notation =,
when we de ne a new object).
Because D :HS ! HC, the adjoint operator is de ned in terms of the inner
products
~ ~
(3.17) (DW;U) =(W; D U) ;
HC HS
which can be translated into the formal inner products as
~ ~
(3.18) [DW;CU] =[W;S D U] :
HC HS
y
The formal adjoint D of D is de ned to be the adjoint in the formal inner product
y
~ ~
(3.19) [W; D CU] =[W;S D U] :
HS HS
y
~
This relationship must be true for all W and U. Therefore D C =S D or D =
¡1 y
S D C; and the discrete analog of the operator grad can be represented as
¡1 y
(3.20) GRAD =¡D =¡S D C:
¡1
Because the operatorS is banded, its inverseS is full on nonorthogonal grids,
and it is not possible to derive explicit formulas for the components of the operator
GRAD. Consequently, GRAD has a nonlocal stencil.804 J. HYMAN AND M. SHASHKOV
The discrete ux,
¡1 y
~
W =¡GRADU =S D CU;
is obtained by solving the banded linear system (recall thatC,S, and D are local
operators)
y
~
(3.21) SW = D CU;
y
where the righthand side, F =(FS ;FS )= D CU,is
FS =¡S (U ¡U );
i;j+1=2 i;j+1=2 i+1=2;j+1=2 i¡1=2;j+1=2
(3.22)
FS =¡S (U ¡U ):
i+1=2;j i+1=2;j i+1=2;j+1=2 i+1=2;j¡1=2
The discrete operatorS is symmetric positive de nite and can be represented as
matrix with ve nonzero elements in each row (see (2.16) and Figure 2.7).
To prove that GRADU is zero if and only ifU is constant, note that (3.22) implies
y ¡1 y
that ifU is a constant, thenD MU = 0 and therefore GRADU =S D MU =0.
Conversely, if we assume GRADU = 0, then (3.20) and the fact thatS is positive
de nite imply
y
(3.23) D MU =0:
This and (3.22) then give
U ¡U =0; i=1;:::;M; j =1;:::;N¡ 1;
(i+1=2;j+1=2) (i¡1=2;j+1=2)
U ¡U =0; i=1;:::;M¡ 1;j =1;:::;N;
(i+1=2;j+1=2) (i+1=2;j¡1=2)
which implies that U is a constant. Therefore the null space of the discrete operator
GRAD is composed only of the constant functions.
3.2.2. Operator DIV. The operator DIV is de ned as the negative adjoint to
0
the natural operator GRAD. In the subspace of scalar functions, , where u(x;y)=
H
0; (x;y)2@V , the boundary term in integral identity (1.1) is zero, and therefore
Z Z
~ ~
(3.24) (W; grad u)dV =¡ u divWdV:
V V
That is, in this subspace div is the negative adjoint of grad in the sense of
Z Z
~ ~ ~ ~
(3.25) (u;v) = uvdV and (A;B) = (A;B)dV :
0
H
H
V V
0
The discrete adjoint operator DIV :HL!HN is de ned as the negative adjoint
0
of the discrete natural operator GRAD :HN!HL,
def
(3.26) DIV = ¡GRAD :
Using the connections between the formal and natural inner products,
¡1 y
(3.27) DIV =¡N GRAD L :DISCRETE ORTHOGONAL DECOMPOSITION 805
(i,j+1)
(i,j)
(i1,j)
(i+1,j)
(i,j1)
h
AL x AL
0
Fig. 3.1. Stencil for the operator DIV =¡GRAD :HL!HN.
y
DIV is local becauseN is diagonal, and both GRAD andL are local. It is easy to
see that
AL AL AL AL
i+1=2;j i¡1=2;j i;j+1=2 i;j¡1=2
y
~
¡GRAD A = ¡ + ¡ :
l l l l
i+1=2;j i¡1=2;j i;j+1=2 i;j¡1=2
The stencil for DIV at the interior nodes (shown in Figure 3.1) is obtained by
combining this formula with the stencil for operators L ;L ;L , and L de ned
11 12 21 22
in (2.22).
~
3.2.3. Operator CURL. In the subspace of vectorsA, where the surface inte
gral in (1.2) on the righthand side vanishes, we have
Z Z
~ ~ ~ ~
(3.28) (curlA;B)dV = (A; curlB)dV :
V V
That is, in this subspace of vector functions, curl is selfadjoint,
(3.29) curl = curl ;
in the inner product
Z
~ ~ ~ ~
(3.30) (A;B) = (A;B)dV :
H
V
~ ~
In the discrete case, for A2HL, the vector CURLA2HS, and we de ne the
discrete adjoint operator CURL :HS!HL as the adjoint to the discrete natural
operator CURL :HL!HS by
def
(3.31) CURL = CURL :
That is,
~ ~ ~ ~
(3.32) (CURLA;B) =(A; CURLB) :
HS HL
We can express CURL as
¡1 y
(3.33) CURL =L CURL S806 J. HYMAN AND M. SHASHKOV
and see that although CURL is a local operator, the operator CURL is nonlocal.
~ ~
We can determine C = CURLB by solving the system of linear equations
y
~ ~
(3.34) L C = CURL S B;
y
with local operatorsL and CURL S .
~ ~
When B =(0; 0;BS ), it easy to prove that the condition CURLB = 0 implies
thatBS = const. The proof is similar to one for GRAD.
i;j
3.2.4. Properties of adjoint discrete operators. The vector analysis theo
rems for adjoint operators are proved in [17]. In this paper we use two theorems:
~ ~ ~ ~ ~
DIVA = 0 if and only if A = CURLB, where A2HL and B2HS;
~ ~ ~
CURLA = 0 if and only if A = GRADU, where A2HS and U2HC.
4. Orthogonal decomposition of discrete vector functions.
4.1. Continuous case. The orthogonal decomposition theorem for continuous
operators states that we can nd the unique vector ~a, which satis es the system of
equations
(4.1) div~a = ; (x;y;z)2V;
(4.2) curl~a =!; ~ (x;y;z)2V;
and one of the boundary conditions,
(4.3) (~a;~n)= ; (x;y;z)2@V ;
where@V is the boundary of V ,~n is the outward unit normal to@V,or
(4.4) [~a ~n]=~ ; (x;y;z)2@V ;
where the tangential components of vector~a are given on the boundary.
In addition the vector !~ must satisfy
(4.5) div!~ =0
because div!~ = div curl~a 0; and there are additional compatibility conditions
that depend on the boundary conditions (see, for example, [12, 19, 21]). When the
normal components of the vector are on the boundary, the additional compatibility
condition needed is
Z I
(4.6) dV = dS:
V @V
When the tangential components of the vector are on the boundary, the compatibility
condition is
Z I
~
(4.7) (!; ~ ~n)dS = (~ ; l)dl;
S @S
~
where S is the part of @V , spanned by the contour @S, and l is the unit tangential
vector to@S.
Moreover, the vector~a can be represented in the form
(4.8) ~a =~a +~a ;
1 2DISCRETE ORTHOGONAL DECOMPOSITION 807
where
(4.9) div~a = ; curl~a =0;
1 1
(4.10) div~a =0; curl~a =!: ~
2 2
From these equations, we can conclude that there is a scalar function ’ and a vector
~
function b such that
~
(4.11) ~a = grad’; ~a = curlb;
1 2
and the vector~a can be decomposed into two orthogonal parts:
~
(4.12) ~a = grad’ + curlb:
For the case where the normal component of~a is given on the boundary, we have
(4.13) div grad’ = ; (x;y;z)2V;
(4.14) (grad’;~n)= ; (x;y;z)2@V;
and
~
(4.15) curl curlb =!; ~ (x;y;z)2V;
~
(4.16) (curlb;~n)=0; (x;y;z)2@V:
The representation (4.12) is called the orthogonal decomposition of~a because the
vector functions~a and~a are orthogonal to each other, (~a;~a ) =0; in the sense
1 2 1 2
H
of the inner product. In fact,
Z
def
(4.17) (~a;~a ) = (~a;~a )dV
1 2 1 2
H
V
Z
~
(4.18) = grad’; curlb dV
V
Z I
~ ~
= ¡ ’ div curlbdV + ’ (~n; curlb)dS
V @V
=0:
~
This follows from the boundary condition (4.16) and the fact that div curlb = 0 for
~
any vector b.
The Neumann boundary condition (4.14) allows the scalar function ’ to be de
~
termined only up to an arbitrary constant. Note that if b is set equal to zero on the
boundary, then it automatically satis es the Dirichlettype boundary condition (4.16)
~
for b.
~
When the tangential components of~a are given on the boundary,’ andb satisfy
the equations
(4.19) div grad’ = ; (x;y;z)2V;
~
(4.20) [grad’ ~n]= 0; (x;y;z)2@V;
and
~
(4.21) curl curlb =!; ~ (x;y;z)2V;
~
(4.22) [curlb ~n]=~ ; (x;y;z)2@V:808 J. HYMAN AND M. SHASHKOV
Here again, the vector functions~a and~a are orthogonal to each other:
1 2
Z
~
(4.23) (~a;~a ) = grad’; curlb dV
1 2
H
V
Z I
~ ~
= curl grad’; b dV¡ b; [grad’ ~n]
V @V
=0:
This follows from the boundary condition (4.20) and the fact curl grad’ = 0 for any
scalar function ’.
We can take the function’ equal to zero on the boundary to satisfy the Dirichlet
~
boundary condition (4.20). For the Neumann boundary condition (4.22) forb, we can
~
determine b up to the gradient of an arbitrary scalar function.
In this paper, we consider only the 2D case when the vector~a depends on (x;y)
and lies in (x;y) plane. For simplicity, we will call these 2D vectors.
For the 2D vector~a, the vector !~ = curl~a has only the ! component, which
z
we, for simplicity, will denote by !. Also the rst compatibility condition in (4.5),
div!~ = 0, is always satis ed because vector ! is independent of z. In the 2D case,
z
vector [~a ~n]=~ in the boundary condition (4.4) has a single component and can be
considered a scalar, which we denote by . The compatibility condition (4.7) reduces
to the case where S is a 2D domain, and @S is the boundary contour. Note that
(4.15) and (4.21) are scalar equations in the 2D case.
We will now consider the orthogonal decomposition of discrete vector functions for
both the case where the normal component of the vector for discrete vector functions
from the spaceHS is given or where the tangential component of the vector for
discrete vector functions from the spaceHL is given.
4.2. Orthogonal decomposition of vector functions inHS. We rst con
~ ~
sider the orthogonal decomposition of 2D vector functions A2HS, where A =
(AS ; AS ; 0).
The discrete orthogonal decomposition states that we can nd a vector function
~
A2HS, satisfying the conditions
i=1;:::;M¡ 1;
~
(4.24) DIVA = ;
i+1=2;j+1=2
j =1;:::;N¡ 1;
i+1=2;j+1=2
i=2;:::;M¡ 1;
~
(4.25) CURLA =! ;
i;j
j =2;:::;N¡ 1;
i;j
AS = L ;AS = R ;j =1;:::;N¡ 1;
1;j+1=2 j+1=2 M;j+1=2 j+1=2
(4.26)
AS = B ;AS = T ;i=1;:::;M¡ 1:
i+1=2;1 i+1=2 i+1=2;N i+1=2
~ ~
That is, the equation DIVA = is satis ed in all the cells, the equation CURLA =
~
! is satis ed at all the internal nodes, and the components of vector A on the boundary
faces are given.
The discrete compatibility condition (and analog of (4.6)) for and is
M¡1N¡1 N¡1
X X X
¡
VC = R S ¡ L S
i+1=2;j+1=2 i+1=2;j+1=2 j+1=2 M;j+1=2 j+1=2 1;j+1=2
i=1 j=1 j=1
M¡1
X
¡
(4.27) + T S ¡ B S :
i+1=2 i+1=2;N i+1=2 i+1=2;1
i=1DISCRETE ORTHOGONAL DECOMPOSITION 809
The discrete analog of the condition div!~ = 0 is always satis ed because the vector
! has only a z component, which depends only on (x;y).
Theorem 4.1. The solution to (4.24){(4.27) is unique.
~ ~
Proof. We prove that the solution is unique by contradiction. AssumeA andA
1 2
~ ~
are two di erent solutions, then ~" =A ¡A satis es
1 2
(4.28) DIV~" =0;
(4.29) CURL~" =0;
"S =0;"S =0;j =1;:::;N¡ 1;
1;j+1=2 M;j+1=2
(4.30)
"S =0;"S =0;i=1;:::;M¡ 1:
i+1=2;1 i+1=2;N
From (4.29), we can conclude
(4.31) ~" = GRAD’
on the internal faces. Normal components of vector~" are equal to zero on the bound
ary (4.30). We know that for the subspace of vector functions with zero normal
components on the boundary, the discrete divergence is the negative adjoint of the
gradient
(4.32) DIV =¡GRAD ;
or
(4.33) (DIV~";’) +(GRAD’;~") =0:
HC HS
Using (4.28), (4.31), and (4.33), we can conclude
(GRAD’; GRAD’) =0;
HS
which implies
GRAD’=0:
Therefore ~" = 0 both on the internal faces (4.31), and on the boundary faces (4.30).
~ ~
Hence A =A , and the solution is unique.
1 2
Theorem 4.2. There exists a solution of problem (4.24){(4.27).
Proof. We will nd the solution of (4.24){(4.27) in the form
~ ~ ~
(4.34) A =G +C;
~ ~
where the vectors G and C satisfy the equations
~
(4.35) CURLG =0;
~
(4.36) DIVG = ;
GS = L ;GS = R ;j =1;:::;N¡ 1;
1;j+1=2 j+1=2 M;j+1=2 j+1=2
(4.37)
GS = B ;GS = T ;i=1;:::;M¡1;
i+1=2;1 i+1=2 i+1=2;N i+1=2
~
(4.38) DIVC =0;
~
(4.39) CURLC =!;
CS =0;CS =0;j =1;:::;N¡ 1;
1;j+1=2 M;j+1=2
(4.40)
CS =0;CS =0;i=1;:::;M¡ 1:
i+1=2;1 i+1=2;N810 J. HYMAN AND M. SHASHKOV
~ ~ ~ ~ ~
By linearity, if the problems forG andC have a solutions, thenA =G +C is the
solution of our original problem.
First we note that at the internal faces (4.35) can be replaced by
~
(4.41) G = GRAD’:
Equations (4.36) and (4.41) with boundary conditions (4.37) give the discrete Poisson
~
equation with Neumann boundary conditions and can be solved by eliminatingG and
~
moving all known values ofG on the boundary to the righthand side of (4.36). From a
formal point of view, the transformed equations are equivalent to a problem with zero
Neumann boundary conditions and a modi ed righthand side, where the operator
DIV is de ned on the subspace of vector functions with zero normal components on
the boundary
(4.42) DIV GRAD’ = :e
Because DIV = ¡GRAD on the subspace of vector functions with zero normal
components on the boundary, DIV GRAD is a selfadjoint operator. The number of
linear equations for ’ in system (4.42) is equal to the number of unknowns (and
i;j
equal to number of cells), (M¡ 1) (N¡ 1).
Equation (4.42) also can be written as
~
~
(4.43) DIVG=~ ;
~
~
(4.44) G = GRAD’;
~ ~
GS =0; GS =0;j =1;:::;N¡ 1;
1;j+1=2 M;j+1=2
(4.45)
~ ~
=0; GS =0;i=1;:::;M¡ 1;
GS
i+1=2;1 i+1=2;N
~
~ ~
where G G on all the internal faces. The compatibility condition (4.27) is
M¡1N¡1
X X
(4.46) e VC =0:
i+1=2;j+1=2 i+1=2;j+1=2
i=1 j=1
When the number of unknowns is equal to the number of equations, then a system
of linear equations has a solution if and only if the righthand side is orthogonal to
all the solutions of the homogeneous adjoint equation [38].
The homogeneous adjoint problem
~
(4.47) DIVF =0;
~
(4.48) F = GRADˆ;
FS =0;FS =0;j =1;:::;N¡ 1;
1;j+1=2 M;j+1=2
(4.49)
FS =0;FS =0;i=1;:::;M¡ 1;
i+1=2;1 i+1=2;N
hasˆ = const: as a solution. To prove this is the only solution, notice that ifˆ is the
solution of homogeneous system, then DIV GRADˆ = 0. Because DIV =¡GRAD ,
we have
(4.50) 0 = (DIV GRADˆ;ˆ) =(GRADˆ; GRADˆ) ;
HC HS
and therefore GRADˆ =0; which requires ˆ = const.DISCRETE ORTHOGONAL DECOMPOSITION 811
The righthand side e of (4.43) is orthogonal to the constant function, which is
equal to one in all cells; I =I = const:;
i;j
M¡1N¡1
X X
(4.51) ( ;eI ) = constant e VC =0;
HC i+1=2;j+1=2 i+1=2;j+1=2
i=1 j=1
using compatibility conditions (4.46). Therefore there is a solution to our original
~
problem forG. Furthermore, as in the continuous case,’ is de ned up to a constant,
~
and because GRADI =0; the vector G = GRAD’ is unique.
~
Next we show that there is a unique vector C satisfying (4.38){(4.40).
~ ~ ~
From condition (4.38) we know that C = CURLB on all faces. Because C is a
~
2D vector, the vector B has a single component,BL ; then
BL ¡BL
i;j+1 i;j
(4.52) CS = ;
i;j+1=2
l
i;j+1=2
BL ¡BL
i+1;j i;j
(4.53) CS =¡ ;
i+1=2;j
l
i+1=2;j
and we can satisfy the boundary conditions (4.40) by choosing BL equal to zero
i;j
on the boundary.
~ ~
The equations for C are equivalent to the problem for the vector B;
~
(4.54) CURL CURLB =!;
BL =0;BL =0;j =1;:::;N;
1;j M;j
(4.55)
BL =0;BL =0;i=1;:::;M:
i;1 i;N
~
If we knowBL , then C is determined by (4.52), (4.53).
i;j
To show that the solution of (4.54), (4.55) is unique, note that for vector functions
with zero boundary conditions CURL = CURL , and therefore
~ ~ ~ ~
(4.56) (CURL CURLB; B) = (CURLB; CURLB) 0;
HL HS
or
CURL CURL 0:
If
~ ~ ~ ~
(4.57) (CURL CURLB; B) = (CURLB; CURLB) =0;
HL HS
~ ~
then CURLB =0; and because B has only one component, BL , BL = const.
i;j
Also, because of BL = 0 on the boundary, then BL = 0 everywhere. This
i;j i;j
implies that the operator CURL CURL is positive; CURL CURL> 0.
Positive operators are invertible in a nitedimensional space. Therefore CURL
CURL has an inverse, and the system (4.54), (4.55) has a unique solution.
~
Theorem 4.3. GRAD’ is orthogonal to CURLB.
~
Proof. Because of the zero boundary conditions for B,
~ ~
(4.58) GRAD’; CURLB = CURL GRAD’; B ;
HS HL
and since CURL GRAD’ = 0 for any ’,wehave
~
(4.59) GRAD’; CURLB =0;
HS
~
proving that GRAD’ is orthogonal to CURLB.812 J. HYMAN AND M. SHASHKOV
4.3. Orthogonal decomposition of vector functions in HL. We now
~ ~
consider the orthogonal decomposition of vector functions A2HL, where A =
(AL ; AL ; 0), and its tangential components AL , AL are given on correspond
ing parts of the boundary.
~
The goal is to a nd vector function A2HL, satisfying the conditions
i=1;:::;M¡ 1;
~
(4.60) DIVA = ;
i;j
j =1;:::;N¡ 1;
i;j
i=1;:::;M¡ 1;
~
(4.61) CURLA =! ;
i+1=2;j+1=2
j =1;:::;N¡ 1;
i+1=2;j+1=2
AL = L ; AL = R ;j =1;:::;N¡ 1;
1;j+1=2 j+1=2 M;j+1=2 j+1=2
(4.62)
AL = B ; AL = T ;i=1;:::;M¡ 1:
i+1=2;1 i+1=2 i+1=2;N i+1=2
~
That is, the equation DIVA = is satis ed at all the internal nodes, the equation
~
CURLA =! is satis ed in all the cells, and the tangential components of the vector
~
A are given on the boundary edges. The discrete compatibility condition for ! and
’s,
M¡1N¡1 N¡1
X X X
¡
! VC = R S ¡ L S
i+1=2;j+1=2 i+1=2;j+1=2 j+1=2 M;j+1=2 j+1=2 1;j+1=2
i=1 j=1 j=1
M¡1
X
¡
(4.63) ¡ T S ¡ B S ;
i+1=2 i+1=2;N i+1=2 i+1=2;1
i=1
also has to be satis ed.
Theorem 4.4. The solution to (4.60){(4.62) is unique.
~ ~
Proof. We prove the uniqueness of the solution contradiction. If A and A are
1 2
~ ~
two di erent solutions, then ~" =A ¡A satis es
1 2
(4.64) DIV~" =0;
(4.65) CURL~" =0;
"L =0; "L =0;j =1;:::;N¡ 1;
1;j+1=2 M;j+1=2
(4.66)
"L =0; "L =0;i=1;:::;M¡ 1:
i+1=2;1 i+1=2;N
From (4.64), we can conclude
~
(4.67) ~" = CURLB
on the internal edges.
The tangential components of the vector~" are zero on the boundary by (4.66). In
the subspace of vector functions with zero tangential components on the boundary,
the discrete operators CURL and CURL are adjoint to each other, CURL = CURL ;
or
~ ~
(4.68) (CURL~"; B) +(CURLB;~") =0:
HS HL
~ ~
Using (4.65) and (4.67), we conclude that (CURLB; CURLB) =0; and therefore
HL
~ ~
CURLB =0: Because~" = CURLB = 0 on the internal edges, and it is equal to zero
on the boundary (4.66), then~" =0; and the solution is unique.
Theorem 4.5. There exists the solution of problem (4.60){(4.62).DISCRETE ORTHOGONAL DECOMPOSITION 813
Proof. We will nd the solution of (4.60){(4.62) in the form
~ ~ ~
(4.69) A =G +C;
~
where the vector G satis es the equations
~
(4.70) CURLG =0;
~
(4.71) DIVG = ;
GL =0; GL =0;j =1;:::;N¡ 1;
1;j M;j
(4.72)
GL =0; GL =0;i=1;:::;M¡ 1;
i;1 i;N
~
and the vector C satis es the equations
~
(4.73) DIVC =0;X
~
(4.74) CURLC =!;
CL = L ;CL = R ;j =1;:::;N¡ 1;
1;j+1=2 j+1=2 M;j+1=2 j+1=2
(4.75)
CL = B ;CL = T ;i=1;:::;M¡ 1:
i+1=2;1 i+1=2 i+1=2;N i+1=2
~
We rst consider the problem for G and replace (4.70) by the equation
~
G = GRAD’:
Because the components GL ;GL of the vector GRAD’ are equal to
i;j i;j
’ ¡’i;j ’ ¡’i;j
i+1;j i;j+1
(4.76) GL = ; GL = ;
i+1=2;j i;j+1=2
l l
i+1=2;j i;j+1=2
the boundary conditions (4.72) will be satis ed if we choose ’ = 0 on the boundary.
i;j
Therefore, instead of the original problem, we can consider
~
(4.77) DIVG = ;
~
(4.78) G = GRAD;’
’ =0;’ =0;j =1;:::;N;
1;j M;j
(4.79)
’ =0;’ =0;i=1;:::;M:
i;1 i;N
This is a discrete Poisson’s equation with Dirichlet boundary conditions. It can easily
be shown that the solution is unique using the fact that DIV and GRAD are negative
adjoint to each other in the subspace of discrete scalar functions that are zero on the
boundary. The proof is almost identical to one for the Dirichlet problem in the case
ofHS vectors.
~
Now let us consider problem (4.70){(4.72) for the vectorC. The condition (4.73)
~ ~ ~
implies thatC = CURLB on the internal edges. Because the vectorC is a 2D vector,
~
the vector B has only one component,BS .
This Neumann boundary problem is selfadjoint, and the compatibility condition
(4.63) guarantees that the righthand side of the nonhomogeneous equation is orthog
onal to all solutions (constants) of the adjoint homogeneous equation. The proof
~
that this problem has a unique (up to constant) solution and that the vector C is
unique follows from a simple modi cation of the proof for the HS case with Neumann
boundary conditions.
~
Theorem 4.6. GRAD’ is orthogonal to CURLB.814 J. HYMAN AND M. SHASHKOV
0.6
0.4
0.2
0
0.2
0.4
0.6
0.6 0.4 0.2 0 0.2 0.4 0.6
Fig. 5.1. Continuous spiral vector eld ~a.
Proof. Because of the zero boundary conditions for the scalar function ’,
~ ~
(4.80) GRAD’; CURLB = ’; DIV CURLB ;
HL HN
~ ~
and because DIV CURLB = 0 for any B,
~
(4.81) GRAD’; CURLB =0:
HL
~
Hence GRAD’ and CURLB are orthogonal.
5. Numerical example. We will illustrate these theorems on the orthogonal
decomposition of a spiral vector eld. The continuous spiral vector eld ~a given by
its Cartesian components (a (x;y);a (x;y)),
x y
(5.1) a (x;y)=x¡y; a (x;y)=x +y;
x y
is shown in Figure 5.1 in the square [¡0:5; 0:5] [¡0:5; 0:5].
This vector eld satis es
(5.2) div~a=2; curl~a=2;
and the values of the normal components of the vector on the boundary can be
obtained from the exact solution.
We rst consider approximation of problem (5.2) in space HS:
~ ~
(5.3) DIVA=2; CURLA=2;
~ ~
whereA2HS and the components ofA are given on the boundary, and then solve for
~ ~
the orthogonal decomposition of A = GRAD’ + CURLB, satisfying (4.35){(4.40).
We solve (5.3) on the smooth grid (shown in Fig. 5.2 (a)) obtained by mapping
the uniform grid in square [¡0:5; 0:5] [¡0:5; 0:5] in the space ( ; ) into the same
square in computational space x( ; );y( ; ):
x( ; )= +0:1 sin(2 ) sin(2 );y( ; )= +0:1 sin(2 ) sin(2 ):
The numerical solution of 5.3 for M =N = 17 is shown in Figure 5.2 (b).DISCRETE ORTHOGONAL DECOMPOSITION 815
0.6
0.6
0.4
0.4
0.2 0.2
0 0
0.2 0.2
0.4
0.4
0.6
0.6
0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 0.2 0.4 0.6
~
(a) Computational grid (b) Discrete vector eld A
Fig. 5.2. (a) The smooth computational grid, used in solution of (5.3). (b) The recovered
~
(reconstructed) discrete vector eld A.
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0.2 0.2
0.4 0.4
0.6 0.6
0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 0.2 0.4 0.6
~
(a) Discrete vector eld GRAD’ (b) Discrete vector eld CURL B
~ ~
Fig. 5.3. The orthogonal decomposition of vector A = GRAD’ + CURLB satisfying (5.3). (a)
~
Vector function GRAD’, M =N =17. (b) Vector function CURLB, M =N =17.
To solve discrete equations we use the multigridpreconditioned conjugategradient
method described in [25].
~
The results of orthogonal decomposition, vector functions GRAD’ and CURLB,
are shown in Figure 5.3 (a) and (b).
Table 5.1 veri es the secondorder convergence rate for the approximate vector
~
eld A to the exact solution in the max norm
E E
(5.4) ˆ = maxfmaxjAS ¡AS j; maxjAS ¡AS jg;
i;j+1=2 i;j+1=2 i+1=2;j i+1=2;j
i;j i;j
E E
where AS ;AS are the approximate solutions, and AS ;AS
i;j+1=2 i+1=2;j
i;j+1=2
are the projections of the exact spiral vector eld, de ned by
i+1=2;j
E x
(5.5) AS =n 0:5[a (x ;y )+a (x ;y )]
x i;j+1 i;j+1 x i;j i;j
i;j+1=2 i;j+1=2
y
+n 0:5[a (x ;y )+a (x ;y )] ;
y i;j+1 i;j+1 y i;j i;j
i;j+1=2816 J. HYMAN AND M. SHASHKOV
Table 5.1
~
Convergence rate for the approxiamte vector eld A to the exact solution
M =N 17 33 65
max norm 1.11E2 3.05E3 7.85E4
q 1.86 1.95 
max
E x
(5.6) AS =n 0:5[a (x ;y )+a (x ;y )]
x i+1;j i+1;j x i;j i;j
i+1=2;j i+1=2;j
y
+n 0:5[a (x ;y )+a (x ;y )] ;
y i+1;j i+1;j y i;j i;j
i+1=2;j
¡ ¡
y y
x x
where n ;n and n ;n are Cartesian components of
i;j+1=2 i;j+1=2 i+1=2;j i+1=2;j
the unit normal vectors to the facesS andS .
i;j+1=2 i+1=2;j
6. Discussion. The orthogonal decomposition theorem of a vector eld is one
of the most important theorems of vector analysis. In this paper, we proved that the
new mimetic discrete approximations of the divergence, gradient, and curl derived
using the SOM automatically satisfy discrete analogs of the theorem for vectors from
spacesHL of discrete vector functions de ned by their orthogonal projections onto
directions of the edges of the cell andHS of discrete vector functions de ned by their
orthogonal projections onto directions perpendicular to face of the cell. That is, any
~
vector A2HS can be represented uniquely as
~ ~
(6.1) A = GRAD’ + CURLB;
~ ~
where ’2HC and B2HL, and similarly any vector A2HL can be presented as
~ ~
(6.2) A = GRAD’ + CURLB;
~
where ’2HN and B2HS.
The approximations for div, grad, and curl described in detail in [16, 17] in
clude the natural discrete divergence, gradient, and curl operators based on coordinate
invariant de nitions, such as Gauss’s theorem, for the divergence, and the formal ad
joints of these operators. The operators have compatible domains and ranges and can
be combined to form all the compound operators div grad, grad div, and curl curl.
By construction, all of these operators satisfy discrete analogs of the integral identities
satis ed by the di erential operators. Furthermore, all the discrete operators de ned
by this selfconsistent approach satisfy analogs of the major theorems of vector anal
ysis relating the di erential operators. These discrete operators have proven to be
e ective in solving the heat conduction equation [36], [37], [18], Maxwell equations
[9], [1], and the equations of magnetic eld di usion [34]. In our continuing quest
to create a discrete analog of vector analysis on logically rectangular, nonorthogonal,
nonsmooth grids, we are comparing the accuracy and stability of the SOM meth
ods with the commonly used FDM and niteelement methods and are investigating
higherorder methods on nonuniform grids. We will use the theorems in this paper to
investigate the stability and accuracy of SOM methods using the energy method in
an approach similar to what has been used in [13], [30], [31] for rectangular grids.
These new methods have great potential for solving complex, nonlinear PDEs on
nonuniform grids and preserving the conservation laws and geometric properties of
the underlying physical problem.
Acknowledgments. The authors are thankful to J. Dendy, J. Dukowicz, L.
Margolin, J. Morel, S. Steinberg, B. Swartz, and R. Nicolaides for many fruitful
discussions.DISCRETE ORTHOGONAL DECOMPOSITION 817
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