Optimal uniform convergence analysis of mixed finite element methods for solving singularly perturbed problems: A unified approach (Q2764441)

An approach to derive uniform convergence results in the numerical solution of singularly perturbed boundary value problems by means of finite element methods on Shishkin type meshes is proposed. To show the generality of the proposed technique several model problems are considered: NEWLINENEWLINENEWLINEIn section 2 the author studies a one dimensional boundary value problem for a singularly perturbed linear fourth order equation. In this case the author takes a Shishkin type mesh, called \(A\)-mesh, which uses \( \sigma = {\mathcal O} ( \varepsilon |\log \varepsilon |) \) as the transition point. In sections 3 and 4 a nonlinear second order problem and a nonsteady semilinear second order problem with Shishkin meshes (\(S\)-mesh) are considered. Finally in section 5 the fourth order problem is reconsidered in an \(S\)-mesh. NEWLINENEWLINENEWLINEIn all cases uniform convergence results in several norms are derived. It is found that, with the proposed technique, the derivation of convergence results with the \(A\)-mesh turns out to be easier than in the case of the \(S\)-mesh.