Finite difference discretizations mimicking fundamental properties of continuum equations.
Mimetic discretizations on polygonal and polyhedral meshes (know-how).
Mimetic finite difference discretizations
An algorithm for deriving mimetic discretizations will be described for
the elliptic equation in the mixed form:
div f = b f = - K grad p
subject to the homogeneous Dirichlet boundary condition.
The solution p and the flux f satisfy the Green formula:
&int&OmegaK-1grad pf dx =
- &int&Omega
div fp dx.
The general steps of the SO methodology are as follows:
define degrees of freedom for physical variables p and f;
equip discrete spaces for p and f with scalar products
[. , .]Q and [. , .]X, respectively;
choose a discrete approximation to the divergence operator,
the prime operator DIV;
derive the discrete approximation of the gradient operator,
the derived operator GRAD, from the discrete Green formula
[f, GRAD p]X =
- [DIV f, p]Q.
Properties of the discretization.
locally conservative
symmetric, i.e. GRAD = -DIV*
can accommodate full diffusion tensor
solution algorithm results in a SPD matrix
exact for linear solutions
second order accurate for cell-centered pressures on unstructured
polygonal (see examples below) and polyhedral meshes
at least first order accurate for fluxes (some minimal solution regularity is required)
may be extended to 3D polyhedral meshes with non-flat faces
can be extended to many other PDEs
Example I: Polygonal meshes
Let p(x, y) = x3 y2 + x cos(xy) sin(x)
be the exact solution and the diffusion tensor be given by
(x+1)2 + y2
-xy
K (x, y) =
-xy
(x+1)2
Let &Omega be the unit square. We consider the sequence of
polygonal median meshes.
The right figure below shows the superconvergence
for both the pressure and the flux.
10-15 PCG iterations for the tolerance 1e-12;
cell centers are given by the mapping
x = &xi +
0.1 sin(2 &pi &xix)
sin(2 &pi &xiy).
Example II: Locally refined meshes
Let p(x, y) , K and &Omega
be as in Example I. We consider the sequence of locally refined quadrilateral meshes.
The right figure below shows the superconvergence
for both the pressure and the flux.
may be used for modeling layered structures in porous media;
compared to other low-order discretizations, the convergence
rate is optimal.
Example III: Locally refined meshes (strong refinement)
Let p(x, y) , K and &Omega
be as in Example I.
We consider the sequence of meshes with strong
refinement in the middle subdomain (see the left figure below).
some of the polygons have a few 180o angles.
Example IV: Non-matching meshes
Let K be piecewise constant and the exact solution be
7/16 - K2 / (12 K1)
+ 2 K2 / (3 K1) y3
y < 0.5
p(x, y) =
y - y4
y &ge 0.5
where K1 and K2 are positive constants.
We consider the sequence of non-matching random meshes.
The second figure below corresponds to K1 = K2 = 1.
The third figure corresponds to K1 = 1 and
K2 = 106.
the mesh aspect ratio varied between 167 and 2024.
Example V: Meshes with non-convex polygons
Let p(x, y) , K and &Omega
be as in Example I. We consider the sequence of random meshes.
Each mesh in the sequence has a fixed amount of non-convex polygons.
about 3% of elements are non-convex.
References.
(2005) F.Brezzi, K.Lipnikov, and V.Simoncini.
A family of mimetic finite difference methods on polygonal and polyhedral meshes.
to appear in M3AS: Mathematical Models and Methods in Applied Sciences.
(2005) F.Brezzi, K.Lipnikov, and M.Shashkov
Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes.
to appear in SIAM J. Numer. Anal.
(2004) K.Lipnikov, J.Morel, and M.Shashkov.
Mimetic finite difference methods for diffusion equations on non-orthogonal
non-conformal meshes. J. Comp. Phys., V.199, pp.589-597.
(2004) Yu. Kuznetsov, K.Lipnikov and M.Shashkov.
Mimetic finite difference method for diffusion equations on polygonal meshes.
to appear in Comput. Geosciences
(2004) M.Berndt, K.Lipnikov, M.Shashkov, M.Wheeler and I.Yotov.
Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals.
to appear in SIAM J. Numer. Anal.
(2001) M.Berndt, D.Moulton, K.Lipnikov, and M.Shashkov.
Convergence of mimetic finite difference discretizations of the diffusion equation. J. Numer. Math., Vol.9, No.4, pp.253-284.
