Wajid Ahmed 14MSM07 Supervisor - Dr. Abdul Hanan Sheikh creator Department of Mathematics And Statistics text Nawabshah Quest 2019 monographic 58p. ABSTRACT In this thesis, the computational efficiency of multigrid solver is discussed with the comparison of existing solvers such as Jacobi-Gauss (JC), Gauss-Seidel (GS), the latter two met hods Conjugate gradient (CG), Generalized minimum residual (GMRES) wit h and without preconditioner. Helmholtz. equation, 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 makes less favorable. In t is work, smoothing parameter in multigrid is tuned to get optimized results in heterogeneous domain . The results are obtained while Inking three different relaxation parameters w = 1, w = 2/3 and w = 1/2 . And different choices of real and imaginary shifts a+bi arc considered for e.g. (0,0),(0,1),(t,1). Results showed t hat better choice of relaxation parameter in smoother is (J) = 2/3. Also multigrid has better convergence with pure imaginary shift i.e. (0,1) compared to rest of choices of shift. Pitfalls in multigrid for indefinite case are discussed along with t he reasons. The proposed techniques help to obtain comparably better results than existing solvers. This fact is validated from presented results, where problem with different gird size 11 and different shift "k" are solved using various.above said, solvers. R/IMS-19 http://tinyurl.com/43vmky9z http://tinyurl.com/43vmky9z