This course will discuss the design of fast iterative solvers for
solving sparse systems of algebraic equations arising from finite
difference or finite element approximation of elliptic partial
differential equations. Our emphasis will be on "optimal" methods where
the work (and cpu time) associated with solution process is
proportional to the dimension of the discretized system. At the heart
of such optimal complexity methods are algebraic or geometric multigrid
cycles applied as preconditioners for Krylov subspace methods.
The following topics will be discussed.
The
Poisson equation. Finite element approximation and properties of the
discretized system.
Ingredients
of multigrid methods for solving the Poisson equation. Smoothing using
Jacobi and Gauss-Seidel iteration. Coarse grid correction. Convergence
theory for two-grid iteration.
Multigrid
preconditioned Conjugate Gradient method for solving the Poisson
equation. Comparison with non-optimal solution methods like incomplete
Cholesky preconditioned CG and with sparse direct solvers.
The
Convection-Diffusion equation. Finite element approximation. Geometric
multigrid for solving discretized convection-diffusion problems.
Multigrid preconditioned GMRES.
Saddle-point
problems. Inf-sup stability. The Stokes equations. Properties of the
discretized system. Multigrid preconditioned MINRES.
The material
will based on chapters 2,4,6 from our recent book: Howard Elman, David
Silvester, Andy Wathen.
Finite Elements and Fast Iterative Solvers: with applications in
incompressible fluid dynamics, Oxford University Press, Oxford, 2005.
ISBN: 978-0-19-852868-5; 0-19-852868-X.
