Volume 55, pp. 706-725, 2022.

A posteriori error estimates for stabilised mixed finite element methods for a nonlinear elliptic problem

María González and Hiram Varela

Abstract

In this paper we propose new adaptive stabilised mixed finite element methods for a nonlinear elliptic boundary value problem of second order in divergence form that appears, among other applications, in magnetostatics. The method is based on a three-field formulation that is augmented with suitable residual least-squares terms arising from the constitutive and equilibrium equations and from the equation that defines the gradient as an additional unknown. We show that the resulting scheme is well posed and obtain optimal error estimates. We also develop an a posteriori error analysis of residual type and derive a simple a posteriori error indicator which is reliable and locally efficient. Finally, we include several numerical experiments that confirm the theoretical results.

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Key words

nonlinear boundary value problem, mixed finite element, stabilisation, a posteriori error estimates, magnetostatics

AMS subject classifications

65N12, 65N15, 65N30, 65N50

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