Algebraic Multigrid

In computer simulation of particular systems (e.g. in physics, biology, economics), partial differential equations (PDE) are solved numerically, mainly by applying Finite Element (FE) or Finite Difference (FD) Analysis. This methods lead to algebraic systems of equations with large, sparse system matrices that have to be solved. Since direct solvers can not exploit the sparsity when the structure of the matrix is unknown (like for unstructured grids), these systems are usually solved with iterative methods. Conventional iterative solvers, like Jacobi, Gauss-Seidel or SOR, need O(N2) work to solve a linear system with N unknowns and thus converge the slower the finer the discretization is; whereas multigrid methods can solve such a linear system with only O(N) work. Hence, they are called optimal or scalable. Because this work can be effectively distributed across a parallel machine, multigrid methods can solve even larger problems on proportionally larger parallel computers in essentially constant time, making them an ideal solver for large-scale scientific simulation. This handout is based on the texts [1],[2],[3].