Direct serendipity and mixed finite elements on polygons and cuboidal hexahedra




Wang, Chuning

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In recent years, there has been a lot of interest in defining finite elements on nonstandard, polytopal elements, i.e., on polygons and polyhedra. Inspired by Arnold and Awanou's serendipity family of finite elements on cuboidal meshes, the dissertation generalizes the construction of direct serendipity and mixed finite elements to general planar, strictly convex polygons. This work also gives an explicit construction of direct serendipity finite elements on convex cuboidal hexahedra. Direct serendipity and mixed finite elements are H1 and H(div) conforming, respectively, and possess optimal order of accuracy for any order. They have a minimal number of degrees of freedom subject to conformity and accuracy constraints. The finite element shape functions are defined to be the full spaces of scalar or vector polynomials plus a space of supplemental functions, of which the choice is not unique. The direct serendipity elements are the precursors of the direct mixed elements in a de Rham complex. The convergence properties of these new families of finite elements are shown under the assumption of shape regularity, as well as some mild restrictions on the choice of supplemental functions. Numerical experiments on various meshes exhibit their performance, especially their potential value in applications where flexible polytopal meshes are advantageous.



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