An observation concerning Ritz-Galerkin methods with indefinite bilinear forms

Abstract

Abstract.Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given.In this note, we would like to discuss existence, uniqueness and estimates over the whole domain for some Ritz-Galerkin methods where the bilinear form satisfies o a Garding type inequality, i.e., it is indefinite in a special way. We shall first illustrate the problem by an example.For simplicity, let £2 be a simply connected convex region in the plane with a polygonal boundary d£2 and consider the Dirichlet problem

Numerical Approximation of Partial Differential Equations

“…An example is provided by Schatz (1974), where the bilinear form is assumed to satisfy the Garding inequality (see 11.1.6). We have limited our analysis to coercive bilinear forms, so that existence and uniqueness of the exact solution is a consequence of the Lax-Milgram lemma.…”

A posteriori error estimators in the finite element method

“…If the domain ~q is not convex, then (as remarked in Schatz) one can show that the assumption holds with p > 0 depending on the maximum interior angle and the nature of the coefficients of L in the neighbourhood of the vertex (see Schatz [14], p. 960-961). There exist positive constants C, kt such that Let us remark immediately that in the case of f~ being a convex domain an application of the Aubin-Nitsche argument shows that the assumption is satisfied with /~ = 1.…”

On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

“…Thus, it only remains to check condition (7), which in general is not an easy task. Nevertheless, the regularity assumption allows us to adapt the approach of Schatz [41] to prove:…”

“…allows us to follow the lines of [41] and derive the existence and stability results for the discrete problem for any ( f, g) ∈ H −1 ( )× L 2 ( ),…”