Contributions to adaptive multigrid methods for the solution of boundary value problems

The main idea of multigrid (MG) is that a boundary value problem is discretized on a sequence of grids with increasing mesh sizes rather than on a single grid. The multigrid idea is based on integration of two principles: error smoothing on fine grids and coarse grid corrections. It is well known that if all components of a multigrid method are chosen correctly for a given problem, MG is fast, optimal G(N), and robust since its convergence rates are independent of the mesh size. Although choosing suitable multigrid components for large classes of problems is well known, it may be (very) difficult to define the right ones in complicated new applications. This is still an art requiring theoretical insight, experience and numerical experiments. <In this thesis, we address some main multigrid difficulties and propose some new developments in the MG field that could have an impact in extending MG applicability. Our first main contribution is the design of efficient MG method to obtain the second solution of Bratu equation which is considered as an example of the difficult nonlinear bifurcation problems. Bratu problem is anonlinear (of exponential type) scalar differential equation containing a parameter C and it has two solutions for some values of C. This difficulty is due to the fact that when solving for the second solution at some parameter values, the linearized discrete system becomes indefinite and hence classical smoothers become unstable. We propose the combination of two Krylovsubspace methods as smoother. The proposed approach enhances the performance of multi grid-based methods for this class of problems. Using the proposed smoother, FMG technique is employed to provide good initial guess on the finest grid and MG-based methods show improved convergence behavior. In the context of adaptive multigrid and finite element methods, we discuss some common algorithmic mistakes and possible incorrect selection of operators that may lead to unexpected results. As a second main contribution, we suggest proper approaches for computing truncation error l’-estimation.Had l’ been accurately estimated, it was exploited using three different techniques to raise the accuracy of the solution as well as to speed up the solution process.