Sergey Korotov, Antti Hannukainen

New computational technologies for reliable computer simulation: ideas and implementation

The present work is devoted to one of the most challenging problems in mathematical modelling and numerical analysis, which also has great importance for many practical applications in industry it reliable verification of the accuracy of approximate solutions obtained via computer simulations. This task is deeply related to the so-called it a posteriori error estimates, giving computable bounds of modelling errors of various types and detecting zones, where such errors are excessively high and some mesh-refinement algorithm should be used.On the base of a model boundary value problem of elliptic type we present new effective ideas and discuss relevant implementation issues on how to obtain two-sided a posteriori error estimates for verifying the overall accuracy of computed solutions in the it global energy norm,item to derive two-sided a posteriori error estimates for the control of local errors in terms of various it quantities of interest,item to develop parallel numerical procedures for computing the above types of a posteriori estimates. Two numerical tests performed with a help of the PDE Toolbox of Matlab and demonstrating the high effectivity of the technologies are presented.