The Cauchy-Riemann equations: discretization by finite elements, fast solution, and a posteriori error estimation

Abstract

In this paper we will concentrate on the numerical solution of the Cauchy-Riemann equations. First we show that these equations bring together the nite element discretizations for the Laplace equation by standard nite elements on the one hand, and by mixed nite element methods on the other. As a consequence, methods for a posteriori error estimation for both nite element methods can derive theirvalidity from each other. Moreover, we show that given a nite element approximation of one of the vectorelds, the missing can be accurately computed in optimal complexity.