Uniform Error Estimation for Convection-Diffusion Problems

Abstract: Let us consider the singularly perturbed model problem $Lu:=-\varepsilon\Delta u-bu_x+c u =f$ with homogeneous Dirichlet boundary conditions on $\Gamma=\partial\Omega$ $u|_\Gamma =0$ on the unit-square $\Omega=(0,1)^2$. Assuming that $b>0$ is of order one, the small perturbation parameter $0<\varepsilon\ll 1$ causes boundary layers in the solution. In order to solve above problem numerically, it is beneficial to resolve these layers. On properly layer-adapted meshes we can apply finite element methods and observe convergence. We will consider standard Galerkin and stabilised FEM applied to above problem. Therein the polynomial order $p$ will be usually greater then two, i.e. we will consider higher-order methods. Most of the analysis presented here is done in the standard energy norm. Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur? We will address this question by looking into a balanced norm. Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly. We will present estimates on the Green's function associated with $L$, that can be used to derive pointwise error estimators.