Calculating the energy-norm FEM-error for Reissner–Mindlin plates without known reference solution

Abstract

The validation of (recently introduced conforming) finite element technologies for the numerical treatment of Reissner–Mindlin plate models requires comparisons with the unknown exact solution. Since mathematical results are often provided for the error in energy norms only it is not sufficient to compare a typical displacement or moment at one point of the domain. Instead of computing a reference solution on a very fine mesh (and then providing a lot of data for the public) we propose the storage of one (problem depending) constant C which then allows an error representation which merely involves known quantities. Based on this approach we could verify convergence rates which were theoretically predicted and give experimental evidence that new adaptive automatic mesh-refining algorithms yield superior approximations. Given any reasonable guess of C (computable from known quantities), our error representation yields an approximation for the unknown error. This establishes a method for a posteriori error control to be employed as a termination criterion.