A remark concerning divergence accuracy order for H(div)-conforming finite element flux approximations. (1st April 2019)

• Record Type: Journal Article
• Authors: Devloo, Philippe R.B.; Farias, Agnaldo M.; Gomes, Sônia M.
• Is Part Of: Computers & mathematics with applications. Volume 77:issue 7(2019)
• Page Start: 1864
• Page End: 1872
• Publication Date: 2019-04-01
• Subjects: Mixed finite elements -- H(div) spaces -- High order convergence rates; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2018.11.019
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: The construction of finite element approximations in H ( div, Ω ) usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region Ω . It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide L 2 -errors with accuracy of order k + 1 for sufficiently smooth flux functions, and of order r + 1 for flux divergence. An example is R T k spaces on quadrilateral meshes, where r = k or k − 1 if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r + n + 1 as desired, for any n ≥ 1 . The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k + n, while keeping the original border fluxes at level k . The case n = 1 has been discussed in the mentioned publication for two particular examples. General stronger enrichment n > 1 shall be analyzed and applied to Darcy's flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme.