A posteriori error estimates and domain decomposition with nonmatching grids
The purpose is to extend the adapive mortar finite element techniques to a nonlinear boundary value problem and to prove that the corresponding residual estimator is asymptotically equivalent to the so-called Babuška-Rheinboldt error estimator [see \textit{I. Babuška} and \textit{W. C. Rheinboldt}, Int. J. Numer. Methods Eng. 12, 1597--1615 (1978; Zbl 0396.65068) and SIAM J. Numer. Anal. 15, 736--754 (1978; Zbl 0398.65069)]. The Petrov-Galerkin finite element method is combined with a domain decomposition method with nonmatching grid. Then the residual estimator for a model semi-linear problem is proposed. The given problem is equivalent formulated to a saddle-point problem which is approximated by a finite element method, in particular, the mortar finite element technique is considered. This mortar method is characterized by introducing a discrete Lagrange multiplier space by a suitable triangulation. For the solution of the saddle-point problem Theorem 1 gives three error estimates. In Section 4, an error estimator for a model problem by a particular choice of the saddle-point problem is discussed. Two numerical results are presented.
scientific article; zbMATH DE number 2170075
scientific article