Immanuel Anjam

Immanuel Anjam, Dirk Pauly

In this paper we present a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations. The error is measured in a combined norm taking into account both the primal and dual variables. We work only on the continuous (often called functional) level and do not suppose any specific properties of numerical methods and discretizations.

Immanuel Anjam, Dirk Pauly

We derive functional a posteriori error equalities and constant free two sided estimates for certain types of partial differential equations. The error is measured in a combined norm which takes into account both the primal and dual variable.

Immanuel Anjam, Dirk Pauly

In this short note we derive, for bounded domains, an upper bound for a Friedrichs type constant in a weighted Friedrichs type inequality. This upper bound generalizes a well known upper bound of the Friedrichs constant. This upper bound is also used to improve an upper bound of a Maxwell type constant for convex domains in $\mathbb{R}^3$. A simple numerical application is also given: we apply the main result in a posteriori error estimation for an elliptic problem.

Immanuel Anjam

In this paper we show how to obtain the exact value of the global error of a conforming mixed approximation of the reaction-convection-diffusion problem. We operate in the framework of functional type a posteriori error control. The error is measured in a combined norm which takes into account both the primal and dual variables. Our main results state that the exact global error value of a conforming mixed approximation is given by a functional which includes only known quantities. The presented error equalities hold with certain restrictions on the reaction coefficient and the convection vector, namely, under the conditions when the solution operators of the corresponding problems are isometries. For the case where these restrictions are not satisfied we derive a two-sided error estimate.

Immanuel Anjam, Jan Valdman

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart-Thomas elements used in discretizations of H(div) spaces and Nedelec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.