Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence. (15th December 2017)

• Record Type: Journal Article
• Authors: Farias, Agnaldo M.; Devloo, Philippe R.B.; Gomes, Sônia M.; de Siqueira, Denise; Castro, Douglas A.
• Is Part Of: Computers & mathematics with applications. Volume 74:issue 12(2017)
• Page Start: 3283
• Page End: 3295
• Publication Date: 2017-12-15
• Subjects: Finite elements -- H(div) spaces -- Mixed formulation -- Approximation space configurations -- Convergence rates; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2017.08.013
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of B D F M k + 1 spaces based on triangles, and of R T k spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k + 2, and matching primal functions of degree k + 1 (higher than the border fluxes, which are kept of degree k ). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors with L 2 -norms, the enriched space configurations give higher convergence rate of order k + 2 for the primal variable, while keeping the order k + 1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k + 1, one unit more than the order k occurring for R T k spaces on this kind of deformed meshes. This fact also holds for A B F k elements, but for them the potential order of accuracy does not improve, keeping order k + 1 .