New a posteriori error indicator in terms of linear functionals for linear elliptic problems

Abstract

This paper is concerned with a posteriori error estimates in terms of goal-oriented quantities for linear elliptic problems. The quantities are presented by locally supported linear functionals that serve as measures of the difference between the exact solution and its approximation in subdomains that can be selected using certain a priori knowledge on the expected properties of the solution. Typically, the evaluation of such type of error indicators is performed with the help of the so-called adjoint problem (see, e.g., [W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin, 2003; R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In: Proc. of ENUMATH'97, 1998, pp. 621–637]). In [S. Korotov, P. Neittaanmäki, and S. Repin, A posteriori error estimation of goal-oriented quantities by the superconvergent patch recovery. J. Numer. Math. (2003) 11, No. 1, 33–59] an approach was suggested, in which the primal and adjoint problems are solved on non-coinciding meshes, and averaging of gradients is used to evaluate the term in the estimator that cannot be computed directly. However, superconvergence requires higher differentiability of the solutions and also regularity (quasi-uniformity) of both primal and dual meshes. In the present paper we propose a new method for computation of the estimator, where we use only superconvergence of the post-processed finite-element solution of the primal problem. At this point our analysis is based on the properties of the least square surface fitting operator investigated by J. Wang [J. Wang, A Superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study (2000) 33, No. 3, 229–243]. We establish asymptotical properties of the new error indicator and verify its efficiency in a series of numerical tests.