Whitney Elements on Sparse Grids

The aim of this work is to generalize the idea of the discretizations on sparse grids to discrete differential forms. The extension to general l-forms in d dimensions includes the well known Whitney elements, as well as H(div; Ω)and H(curl; Ω)-conforming mixed finite elements. The formulation of Maxwell’s equations in terms of differential forms gives a crucial hint how they should be discretized. The focus is on discrete differential forms of lowest order, i.e. Whitney elements. Taking the cue from Lagrangian finite elements on sparse grids, we present the hierarchical decomposition of the Whitney spaces. The tensor product structure and the hierarchical multilevel principle give rise to hierarchical basis for Whitney l-forms in d dimensions. Relying on the hierarchical basis, we define the sparse grid interpolation operator and we prove the commuting diagram property as well as the existence of discrete potentials in sparse grid spaces. The interpolation estimates generalize the known results for Lagrangian finite elements. Approximate interpolation is needed for the Galerkin method for boundary value problems on sparse grids. The combination technique and a two point quadrature rule ensure that similar error estimate as for the exact interpolation hold. Discrete inf-sup conditions are shown theoretically and experimentally 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 existence of stable potentials is a sufficient condition. We prove it in particular cases, completely covering the three dimensional case. Numerical results give evidence for d = 4, too. The results show that discrete differential forms on sparse grids give rise to viable numerical schemes for the discretization of both H(d,Ω)-elliptic variational problems and second order mixed problems. The algorithms involved are presented both in a general form and with particular design solutions. They mirror the two pervasive ideas in the theory of the sparse grids, namely the hierarchical decomposition and the reduction to the one dimensional case via tensor product. We give examples for smooth and no-smooth forms, where we compute numerically the interpolation error on the full grid. Algorithmic details are given for the the multiplication with the mass and the stiffness matrix. The last chapter is confined to the the multigrid scheme and the needed automatic construction of stencils on anisotropic full grids.