Strict, sharp and practical bounds of computed outputs of interest for evolution problems

Abstract

A constant concern both in industry and in research has been the verification of the models used for simulation of physical phenomena. We particularly need to assess the quality of the numerical solutions we get using approximate methods such as the FEM. Effective tools had appeared for thirty years [1] [2], allowing to assess global error (in the energy norm) then local error for quantities of interest which are relevant data for design. For this latter topic, most of the works concern linear problems and yield relatively good bounds of the error for this kind of problem. However, local error estimation for more complex problems has not been mastered yet. Some works dealing with this issue do not yield guaranteed bounds, which is a serious drawback. Others tend to give strict upper and lower bounds but with prohibitive computer ressources. This paper focuses on a method that yields strict bounds for quantities of interest resulting from a finite element analysis of linear evolution problems. The method, developped over the time-space domain, leans on an extraction technique [3] leading to the solution of an adjoint problem. We use dissipation error which is a practical tool developped at the LMT-Cachan for more than ten years [4]. An important feature is the solution of the adjoint problem by means of techniques introducing numerical or analytical functions in space (Partition of Unity Method) and time. Thus, one gets good quality for the bounds on the error with a reasonable numerical cost and without changing the framework of finite element codes. Another aspect of the method is the way to deal with quantities of interest more or less sensitive to history. We take cumulative error effects into account to reach a reliable assessment of the local errors. First results are presented in this paper for linear viscoelasticity problems in 2D. In conclusion, this work which can be extended to non linear problems shows that we can get both good and strict bounds and seems to be therefore a new step forward for the issue of model verification.