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

In the present work, we propose extend an approximation for the second order linear elliptic equation in divergence form with coefficients in L∞ and L1-Data, based on the usual quadratic finite element techniques. We study the convergence with low-regularity solutions only belonging to W1, q0 with (Formula presented.) 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 W1, q0 is then deduced under suitable regularity assumptions on the solution, the coefficients and the L1-Data f.