Iterative solvers for generalized finite element solution of boundary-value problems

Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill-conditioned linear systems that are associated with these methods. In the current study we investigate the performance of a class of iterative solvers for the generalized finite element solution of time-dependent boundary-value problems. A fully implicit time-stepping scheme is used for the time integration in the finite element framework. As enrichment we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared to dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix-vector products are involved in their implementation. Two test examples for boundary-value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods, is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill-conditioned linear systems resulting from the generalised finite element methods for time domain problems.