Interpretation of the algebraic error and algebraic preconditioning as the transformation of the discretization basis

by · Sep 23, 2014 · 1 184 views ·

Jan Papež, Charles University in Prague, Faculty of Mathematics and Physics, Department of Numerical Mathematics, Prague, Czech Republic Joint work with Zdenek Strakoš and Mattia Tani In the finite element method (FEM) used for discretization of partial differential equations, the finite dimensional subspaces are typically generated using locally supported basis functions. The resulting algebraic system matrix is then sparse and the sparsity is presented as an advantage of the FEM. On the other hand, when iterative methods are applied for solving the resulting algebraic problem, an algebraic preconditioning is needed in order to assure fast convergence. In the contribution we first briefly recall, following [1, Chapter 8] (see also the references given there), that any algebraic preconditioning can be interpreted as transformation of the discretization basis and, at the same time, transformation of the inner product in the given Hilbert space. This underlines that discretization and preconditioning are tightly coupled. Second, we present the idea of interpreting the algebraic error as a transformation of the discretization basis. We elaborate on [2, 3] where the transformation of the search basis only is considered and the test functions are unchanged. We discuss the general case with the transformation of both the search and the test basis functions.

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