Analysis of a Quadratic Finite Element Method for Second-Order Linear Elliptic PDE, With Low Regularity Data. (3rd April 2020)

• Record Type: Journal Article
• Authors: Messaoudi, Rachid; Lidouh, Abdeluaab; Seddoug, Belkassem
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 5(2020)
• Page Start: 507
• Page End: 541
• Publication Date: 2020-04-03
• Subjects: Diagonally dominant matrix -- error estimate -- L1-data -- quadratic finite element -- renormalized solution; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2019.1663432
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In the present work, we propose extend an approximation for the second order linear elliptic equation in divergence form with coefficients in L ∞ and L 1 -Data, based on the usual quadratic finite element techniques. We study the convergence with low-regularity solutions only belonging to W 0 1, q with q ∈ [ 1, d d − 1 [ and d ∈ { 2, 3 }, where the class of renormalized solution is considered as limit. Statements and proofs of linear finite elements approximation case in [1 ]; remain valid in our case, and when the Data is a bounded Radon measure, a weaker convergence is obtained. An error estimate in W 0 1, q is then deduced under suitable regularity assumptions on the solution, the coefficients and the L 1 -Data f .