Adaptive Localized Training and Enrichment based on a Residual Localization Strategy
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To tackle parametric partial differential equations with highly heterogeneous coefficients, we propose an adaptive localized basis construction procedure based on both offline training and online enrichment. First, in the offline phase, a set of problem-adpated local basis functions is precomputed. Next, in the online phase, we use a localized residual-based a posteriori error estimator to investigate the accuracy of the reduced solution for any given new parameter. As the error estimator is localized, we can exploit it to adaptively enrich the reduced solution space locally where needed. The approach thus guarantees the accuracy of reduced solutions given any possibly insufficient reduced basis that was constructed during the offline phase. Numerical experiments demonstrate the efficiency of the proposed method.