01930nam a22001217a 4500082001300000100006200013245009700075260003000172300000900202500152000211700004501731856003201776  aR/IMS-19  aWajid Ahmeda14MSM07aSupervisor - Dr. Abdul Hanan Sheikh  aOn The Efficiency OF Multigrid Solver For Shifted Laplace Equation In A Heteroggenous Medium  aNawabshah:bQuest,c2019.  a58p.  aABSTRACT


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.
  aDepartment of Mathematics And Statistics  uhttp://tinyurl.com/43vmky9z