Local Projection Stabilisation on Layer-Adapted Meshes for Convection-Diffusion Problems with Characteristic Layers (Part I and II)

Abstract

AbstractFor a singularly perturbed convection-diffusion problem with exponential and characteristic boundary layers on the unit square a discretisation based on layer-adapted meshes is considered. The standard Galerkin method and the local projection scheme are analysed for a general class of higher order finite elements based on local polynomial spaces lying between \({\mathcal{P}}_{p}\) and \({\mathcal{Q}}_{p}\). We will present two different interpolation operators for these spaces. The first one is based on values at vertices, weighted edge integrals and weighted cell integrals while the second one is based on point values only. The influence of the point distribution on the errors will be studied numerically. We show convergence of order p in the \(\epsilon \)-weighted energy norm for both the Galerkin method and the local projection scheme. Furthermore, the local projection methods provides a supercloseness result of order p in local projection norm.KeywordsInterpolation OperatorLocal ProjectionCharacteristic LayerShishkin MeshHigh Order Finite ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.