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Mathematical Modeling and Analysis High-order Mimetic Finite Difference Methods on Arbitrary Meshes

The mimetic finite difference (MFD) methods mimic important properties of physical and mathematical models. As the result, conservation laws, solution symmetries, and the fundamental identities of the vector and tensor calculus are held for discrete models. The existing MFD methods for solving diffusion-type problems on arbitrary meshes are s cond-orderaccurate for the conservative variable (temperature, pressure, energy, etc.) and only first-orderaccurate for its flux. In many physical simulations such as reactive transport in porous media, compressible flows, etc., the flux accuracy makes significant impact on evolution of conservative quantities. We developed new high-order MFD methods which aresecond-orderaccurate for both conservative variable and its flux [ 1]. These methods are well suited for simulations on arbitrary polygonal meshes. Modelling with arbitrary polygonal meshes has a number of advantages. Such meshes allow to describe accurately small, detailed structures such as tilted layers, irregular inclusions, rugged surfaces and interfaces, etc. The polygonal meshes cover the plane more efficiently than triangular meshes which eventually reduces the number of discrete unknowns without lose of accuracy. The locally refined meshes, used to improve resolution in region of interest, are particular examples of polygonal meshes with degenerate elements. There are a few fundamentally different approaches to increase accuracy of discretization methods. Finite volume and finite difference methods increase stencils of discrete operators which impose severe restrictions on mesh smoothness. These methods are usually applied on smooth meshes and lose accuracy on rough ones. The finite element and spectral element methods increase the number of unknowns inside each element but impose severe restrictions on the shape of admissible mesh elements. To develop new high-order MFD methods, we blend ideas of the finite element [ 4] and the low-order MFD methods [ 2].