The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L p norm and W 1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.

Abstract. We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is re-written as a system of four first-order equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation. For k ≥ 1 a projection of the primal variable error superconverges with order k + 3 while the error itself converges with order k + 1 only. This fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k + 3. We provide numerical experiments that validate our theoretical results.

An adaptive finite element scheme for the advection-reaction-diffusion equation is introduced and analyzed. This scheme is based on a stabilized finite element method combined with a residual error estimator. The estimator is proved to be reliable and efficient. More precisely, global upper and local lower error estimates with constants depending at most on the local mesh Peclet number are proved. The effectiveness of this approach is illustrated by several numerical experiments.

Residual type a posteriori error estimators are introduced in this paper for an advection-diffusion-reaction problem with a Dirac delta source term. The error is measured in an adequately weighted W 1,p -norm. These estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.

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