Mixed finite element approximations based on 3-D hp-adaptive curved meshes with two types of H(div)-conforming spaces

Author(s)

Devloo, P. R. B. ^[1] ; Duran, O. ^[2] ; Gomes, S. M. ^[3] ; Shauer, N. ^[4]

Affiliation

^[1] Univ Estadual Campinas, Fac Engn Civil Arquitetura & Urbanismo, Campinas, SP - Brazil

^[2] Univ Estadual Campinas, Fac Engn Mecan, Campinas, SP - Brazil

^[3] Univ Estadual Campinas, Inst Matemat Estat & Computacao Cient, Campinas, SP - Brazil

^[4] Univ Illinois, Civil & Environm Engn, Urbana, IL 61801 - USA

• Document type: Journal article
• Source: INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING; v. 113, n. 7, p. 1045-1060, FEB 17 2018.
• Web of Science Citations: 4

Abstract

Two stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical-like region, and constant polynomial degree distribution k, show L-2 errors of order k+1 or k+2 for the potential variable, while keeping order k+1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case of hp-adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated. (AU)

• FAPESP's process: 16/13637-5 - Advanced numerical methods: recent developments, analysis, and applications
• Grantee: Sônia Maria Gomes
• Support Opportunities: Research Grants - Meeting - Abroad
• FAPESP's process: 13/21959-4 - Construction of finite element spaces in Hdiv for three- dimensional geometries
• Grantee: Douglas Azevedo Castro
• Support Opportunities: Scholarships in Brazil - Post-Doctoral