The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods

Abstract The support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems that exactly satisfy discrete conservation laws analogous to those satisfied by the continuum system. Thus, the stability of the method is assured, but currently there is no theory that predicts the accuracy of the method on nonuniform grids. In this paper, we numerically investigate how the accuracy, particularly the accuracy of the fluxes, depends on the definition of the inner product for discrete vector fields. We introduce two different discrete inner products, the standard inner product that we have used previously and a new more accurate inner product. The definitions of these inner products are based on interpolation of the fluxes of vector fields. The derivation of the new inner product is closely related to the use of the Piola transform in mixed finite elements. Computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra. From the results of our numerical experiments, we can conclude that using more accurate inner product produces a method with the same order of convergence as the standard inner product, but the constant in error estimate is about three times less. However, the method based on the standard inner product is easier to compute with and less sensitive to grid irregularities, so we recommend its use for rough grids.

[1]  Mikhail Shashkov,et al.  Solving Diffusion Equations with Rough Coefficients in Rough Grids , 1996 .

[2]  M. Shashkov,et al.  A Local Support-Operators Diffusion Discretization Scheme for Quadrilateralr-zMeshes , 1998 .

[3]  M. Shashkov Conservative Finite-Difference Methods on General Grids , 1996 .

[4]  Jim E. Jones,et al.  Control‐volume mixed finite element methods , 1996 .

[5]  Mikhail Shashkov,et al.  Approximation of boundary conditions for mimetic finite-difference methods , 1998 .

[6]  M. Shashkov,et al.  Adjoint operators for the natural discretizations of the divergence gradient and curl on logically rectangular grids , 1997 .

[7]  M. Shashkov,et al.  Support-operator finite-difference algorithms for general elliptic problems , 1995 .

[8]  Len G. Margolin,et al.  Using a Curvilinear Grid to Construct Symmetry-Preserving Discretizations for Lagrangian Gas Dynamics , 1999 .

[9]  M. Shashkov,et al.  The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials , 1997 .

[10]  R. J. MacKinnon,et al.  Analysis of material interface discontinuities and superconvergent fluxes in finite difference theory , 1988 .

[11]  M. Shashkov,et al.  Mimetic Discretizations for Maxwell's Equations , 1999 .

[12]  Patrick M. Knupp,et al.  Fundamentals of Grid Generation , 2020 .

[13]  Todd Arbogast,et al.  Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry , 1998, SIAM J. Sci. Comput..

[14]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[15]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[16]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .