Mixed-Finite-Element Method on Hexahedral Meshes



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In this dissertation, we have three goals. The first goal is to investigate the accuracy behavior due to lumping procedure. The second goal is to investigate the effects of different boundary partitions of macro-cells. The last goal is to numerically verify the error estimate proposed by Kuznetsov in 2011.

In the first part, we propose a new mixed-finite-element approximation method, elaborate its construction and discretization. We, afterwards, propose a new procedure called “coarsening of fluxes” or “lumping” procedure which will impose only one degree of freedom for flux on each quadrilateral face instead of two under admissible conditions.

In the second part, we first derive the optimal boundary partitions for both non-degenerate and degenerate groups. The thesis also introduces two center-based interior partitions which have been widely used nowadays, and we conclude that the optimal boundary partitions discovered in thesis reduce huge amounts of elimination work compared with two center-based interior partitions. We also propose a homogenization procedure which introduce one degree of freedom for solution function in each macro-cell. Finally, the results of numerical experiments demonstrate that with lumping procedure, the errors do not converge to zero, instead they will stay stagnant asymptotically as we refine the mesh.



Mixed-finite-element method, Hexahedral meshes, Lumping procedure