Mixed methods for degenerate elliptic problems and application to fractional Laplacian

¹ Centro de Matemática La Plata, Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calle 50 y 115, La Plata 1900, Prov. de Buenos Aires, Argentina
² IMAS (UBA-CONICET) and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Ciudad Autónoma de Buenos Aires 1428, Argentina
³ Instituto de Matemática (INMABB) - Departamento de Matemática, Universidad Nacional del Sur (UNS)-CONICET, Baha Blanca, Argentina

^* Corresponding author: mec.eugenia@gmail.com

Received: 23 April 2020
Accepted: 8 September 2020

Abstract

We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A₂. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.