Multilevel local defect-correction method for the non-selfadjoint Steklov eigenvalue problems

In this paper, we design a multilevel local defect-correction method to solve the non-selfadjoint Steklov eigenvalue problems. Since the computation work needed for solving the non-selfadjoint Steklov eigenvalue problems increases exponentially as the scale of the problems increase, the main idea of our algorithm is to avoid solving large-scale equations especially large-scale Steklov eigenvalue problems directly. Firstly, we transform the non-selfadjoint Steklov eigenvalue problem into some symmetric boundary value problems defined in a multilevel finite element space sequence, and some small-scale non-selfadjoint Steklov eigenvalue problems defined in a low-dimensional auxiliary subspace. Next, the local defect-correction method is used to solve the symmetric boundary value problems, then the difficulty of solving these symmetric boundary value problems is further reduced by decomposing these large-scale problems into a series of small-scale subproblems. Overall, our algorithm can obtain the optimal error estimates with linear computational complexity, and the conclusions are proved by strict theoretical analysis which are different from the developed conclusions for equations with the Dirichlet boundary conditions.