Complementary Problem Domain Decomposition Method And The Multigrid Method Of Non-symmetric Elliptic Boundary Value Problems

The paper is organized as follows. First, we consider an additive Schwarz algorithm for the solution of Ax 4- F(x) 0, x when coefficient A is an M-matrix and F'(x) 0. By applying the theory of weak regular splitting of matrices to the above considered algorithm, we obtain the weighted max-norrn bound for iterations. Moreover, under the assumption that F(x) is concave, we establish monotone convergence of the considered algorithm. At the same time we give a modified algorithm of the considered algorithm, it's convergence is obtained without any assumption. Second, we consider a multiplicative Schwarz algorithm for the solution of the finite-dimensional nonlinear complementarity problem metioned above and a multiplicative Schwarz algorithm for the solution of Ax - F 0, x (A is an M-matrix). By applying the theories of regular splitting and weak regular splitting of matrices to the two considered algorithms, we obtain the weighted max-norm bounds for iterations too. Finally, We consider a cascadic multigrid method for soiviag a nons.y mmetric elliptic differential equation with boundary conditions. By using a technique to choose iterate numbers on each level, we obtain the convergence. Numerical experiment show that the method is effective.