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Sparsity optimized high order finite element functions for H(div) on simplices. (English) Zbl 1256.65100

The authors investigate the space of vector valued-functions with square-integrable divergence and conforming \(h_p\) finite element discretization for open bounded Lipschitz domains \(\Omega\subset \mathbb R^d\) with \(d=2,3\). A new set of of basis functions for \(H\)(div)-conforming \(h_p\) finite element spaces, which yields an optimal sparsity pattern for system matrices derived from the discretization of some convenient bilinear form, is introduced. The construction of basis functions relies on some known construction principles in the literature which concerns the sparsity of \(H\)(div))-conforming discretizations. More precise, the construction principles are related to the use of Raviart-Thomas elements, mixed-weighted Jacobi polynomials, and Dubiner basis. This construction implies the \(L_2\) orthogonality of the fluxes of the basis.
The proof of sparsity of the mass matrix requires some symbolic computation. The stated finite element basis form a hierarchical set of \(H\)(div)-conforming basis functions and hence are applicable also in general settings on unstructured (curved) simplicial meshes.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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