By Lourenco Beirao da Veiga, Konstantin Lipnikov, Gianmarco Manzini
This publication describes the theoretical and computational points of the mimetic finite distinction approach for a large classification of multidimensional elliptic difficulties, including diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending difficulties. the fashionable mimetic discretization know-how constructed partly via the Authors permits one to unravel those equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The booklet offers a realistic advisor for these scientists and engineers which are drawn to the computational houses of the mimetic finite distinction strategy reminiscent of the accuracy, balance, robustness, and potency. Many examples are supplied to assist the reader to appreciate and enforce this system. This monograph additionally offers the fundamental historical past fabric and describes simple mathematical instruments required to enhance extra the mimetic discretization expertise and to increase it to numerous purposes.
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Extra info for The Mimetic Finite Difference Method for Elliptic Problems
This assumption is quite natural when we deal with partial differential equations with (essential) homogeneous boundary conditions. 7) r u. Jor V' u . 8) r u. Jor (curl u) . v dV. 1. Inhomogeneous boundary conditions can be treated by extending the first-order operators to the boundary, see, for instance, . The DVTC that results from this approach is different. In the mimetic approach, a problem coefficient is combined with a differential operator and the two are discretized simultaneously.
We consider again assumption (HI) from the previous section and modify assumption (H2) as follows. (H2a) The boundary data functions~, gV belong to HI/2(rD) and the dual of H~62(rN), respectively. The load function b belongs to L2(Q). Moreover, we assume that r D has positive measure. 17) is well-posed . 1 holds agam. 3 Advection-diffusion equation in mixed form Many biological and geophysical problems involve transport of the scalar field c (species concentration for mass transfer in porous media or temperature for heat transfer) with the vector field f3.
We denote the sets of mesh nodes v, edges e, faces f, and polyhedra P by Y, 6', §, and 9, respectively. Let 2 be one of these sets. We define the subsets 2(P), 2(f), and 2( e), which are formed by the mesh objects of 2 that are related, respectively, to polyhedron P, face f, and edge e. When the argument has a higher topological dimension, the resulting set 2(P), 2(f), or 2(e) is the collection of mesh objects that belong to the boundary of P, f, and e, respectively. For example, 6'(P) denotes all the edges fonning the boundary of polyhedron P.
The Mimetic Finite Difference Method for Elliptic Problems by Lourenco Beirao da Veiga, Konstantin Lipnikov, Gianmarco Manzini