Analysis of a Quadratic Finite Element Method for Second-Order Linear Elliptic PDE, With Low Regularity Data

Abstract

In the present work, we propose extend an approximation for the second order linear elliptic equation in divergence form with coefficients in $L^{\infty }$ and L¹-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 $q\in [1,\frac{d}{d-1}[$ and $d\in \left\{2,3\right\},$ 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 L¹-Data f.

Keywords:

• Diagonally dominant matrix
• error estimate
• L1-data
• quadratic finite element
• renormalized solution