Multi-grid methods of stable generalized finite element methods for interface problems

The stable generalized finite element method (SGFEM) for interface problems uses simple mesh that is independent of interface curves and is optimally convergent, well conditioned, robust, and free from any penalty parameters. This study proposes multi -grid (MG) -based fast solvers for the SGFEM of interface problems. The difficulty is that (a) the stiffness matrix is not a standard finite element (FE) matrix and contains FE, enrichment, and intersection parts and (b) the trial spaces of fine and coarse meshes do not possess nested structures. To overcome this, we decompose a linear system of SGFEM into two small systems using a Schur complement technique. The size of the first linear system is as big as that of the standard finite element method (FEM). The second system can be efficiently solved using general elimination or iteration methods because the associated matrix is one dimension less than that of the FEM and its condition number is essentially small. The matrix of the first system, which is a FE matrix plus a perturbation, is not a standard FE matrix; however, we determine that the condition number of the matrix is of the same order as that of the FEM. Motivated by this, we successfully develop two MG methods, namely a direct MG method and a MG preconditioned conjugate gradient (MGCG) method, for the first system using the standard FE MG operations. Numerical experiments confirm that the proposed MG methods work quickly and efficiently. Comparisons with other iteration methods, such as conjugate gradients preconditioned by incomplete Cholesky decomposition, also affirm the effectiveness of the proposed MG and MGCG methods.