Proceedings of the 9th Seminar “Programs and Algorithms of Numerical Mathematics”

Preface

Proceedings of the 9th Seminar Programs and Algorithms of Numerical Mathematics     

Guaranteed and locally computable a posteriori error estimate

^** Email: vejchod{at}math.cas.cz A new approach, based on the combination of the equilibratedresidual method and the method of hypercircle, is proposed fora posteriori error estimation. Computer implementation of theequilibrated residual method is fast, but it does not produceguaranteed estimates. On the other hand, the method of hypercircledelivers guaranteed estimates, but it is not fast because itinvolves solving a global linear algebraic system. The combinationof these two methods leads to guaranteed and locally computablea posteriori error estimator. This combined method is appliedto linear elliptic problem in two dimensions with mixed boundaryconditions and non-negative absolute terms.     

Preface

Preface

Preface     

Discrete maximum principle for higher-order finite elements in 1D

We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the -FEM. The DMP holds if a relative length of every element in the mesh is bounded by a value , where is the polynomial degree of the element . The values are calculated for .

     

Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuška (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method.