Mixed methods for degenerate elliptic problems and application to fractional laplacian

Abstract

We analyze the approximation by mixed finite element methods of solutions of equations of the form $-\mbox{div\,} (a\nabla u) = g$, where the coefficient $a=a(x)$ can degenerate going to cero or infinity. First, we extend the classic error analysis to this case provided that the coefficient $a$ belongs to the Muckenhoupt class $A_2$. 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 lowest order, obtaining optimal order error estimates for general regular elements as well as for some particular anisotropic ones 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.