Whitney Elemente auf dünnen Gittern

Abstract

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This work generalize the idea of the discretizations on sparse grids to differential forms. The extension to general l-forms in d dimensions includes the well known Whitney elements, as well as H(div)- and H(curl)- conforming mixed finite elements.

The construction is based on one-dimensional differential forms, related wavelet representations and their tensor products. In addition to the construction of spaces, interpolation estimates are given. They display the typical efficiency of approximations based on sparse grids.

Discrete inf-sup conditions are shown theoreticaly and experimentaly for mixed second order problems. The focus is on the stability of the discretization of the primal and of the dual mixed problem by sparse grid Whitney forms.

The explanation of the involved algorithms received a particular attention, filling a gap in the literature. Details on the multilevel transforms, approximate interpolation operators, mass and stiffness matrix multiplications are given. The construction of general stencils on anisotropic full grids completes the detailed description of the multigrid solver based on semicoarsening.