1 Linear Equation Systems in the Numerical Solution of PDE’s 3 1.1 Examples of PDE’s . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Weak Formulation of Poisson’s Equation . . . . . . . . . . . . 6 1.3 Finite-Difference-Discretization of Poisson’s Equation . . . . . 7 1.4 FD Discretization for Convection-Diffusion . . . . . . . . . . 8 1.5 Irreducible and Diagonal Dominant Matrices . . . ...

Considered are parametrised linear systems which parameters are subject to tolerances. Rump's xed-point iteration method for nding outer and inner approximations of the hull of the solution set is studied and applied to an electrical circuit problem. Interval Gauss-Seidel iteration for parametrised linear systems is introduced and used for improving the enclosures, obtained by the xed-point met...

In the case of convection dominated problems, multi-grid methods require an appropriate smoothing to ensure robustness. As a first approach we discuss a Gauß-Seidel smoothing with a correct numbering of the unknowns and if necessary a special block partitioning. Numerical experiments show that, in the case of general convection directions, the multi-grid algorithms obtained in this way have the...

Multigrid is widely used as an efficient solver for sparse linear systems arising from the discretization of elliptic boundary value problems. Linear relaxation methods like Gauss-Seidel and Red-Black Gauss-Seidel form the principal computational component of multigrid, and thus affect its efficiency. In the context of multigrid, these iterative solvers are executed for a small number of iterat...

In this paper we study a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies. We find that point-wise block-partitioning gives much better results than the classical cell-wise partitioning. Both for the Baumann-Oden and for th...

In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann–Oden and for the symmetric DG method, we give a detailed analysis of the convergence for celland point-wise block-relaxation strategies. We show that, for a suitably constructed two-dimension...

Explicit pre/post conditioning of the large, sparse and non-symmetric system of equations, arising from the discretization of the Dirichlet Poisson’s Boundary Value Problem (BVP) by the Hermite Collocation method is the problem considered herein. Using the 2-cyclic (red-black) structure of the Collocation coefficient matrix, we investigate the eigenvalue distribution of its preconditioned analo...

In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the convergence for celland point-wise block-relaxation strategies. We show that, for a suitably constructed two-dimension...

Good data locality is an important aspect of obtaining scalable performance for multigrid methods. However, locality can be difficult to achieve, especially when working with unstructured grids and sparse matrices whose structure is not known until runtime. Our previous work developed full sparse tiling, a runtime reordering and rescheduling technique for improving locality. We applied full spa...

A simple Gauss-Seidel technique is proposed which exploits the special form of the chemical kinetics equations. Classical Aitken extrapolation is applied to accelerate convergence. The technique is meant for implementation in stii solvers that are used in long range transport air pollution codes using operator splitting. Splitting necessarily gives rise to a great deal of integration restarts. ...