On The Efficiency Of Multigrid Solver For Shifted Laplace Equation In A Heterogeneous Medium

In this paper, the computational efficiency of Multigrid solver is discussed with the comparison of basic solvers such as Gauss-Jacobi, Gauss-Seidel, and Generalized minimum residual (GMRES). The problem considered in work is elliptic partial differential equation, which immediately attracts Multigrid method. Multigrid is reluctant to perform well in heterogeneous geometries. Further shift in Helmholtz problem allows occurrence of negative eigenvalues, making problem indefinite. These two make Multigrid with standard components less favorable. In this work, smoothing parameter in Multigrid is tuned to get optimized results in heterogeneous domain. The results are obtained while taking three different relaxation parameters omega = 1, omega = 2/3 and omega = 1/2 and different choices of real and imaginary shifts a + ib are considered for e.g. (0, 0), (0, 1) and (1, 1). Results showed that better choice of relaxation parameter in smoother is omega = 2/3. Also Multigrid has better convergence with pure imaginary shift i.e. (0, 1) as compared to rest of choices of shift, chosen in this work. The proposed technique helps to obtain comparably better results than existing solvers. This fact is affirmed by presented results, where problem with different gird size n and different shift "k" is experimented.