Hybrid Localized Spectral Decomposition for Multiscale Problems
Alexandre L. Madureira, Marcus V. Sarkis · SIAM Journal on Numerical Analysis · 2021
We consider a finite element method for elliptic equations with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. We assume minimal regularity of the solutions. A space decomposition as in FETI and BDCC induces an embarrassingly parallel preprocessing and leads to a final system of size independent of the coefficients. The resulting solution is in equilibrium, and all PDEs involved are elliptic. One of the problems in the pre-processing step is nonlocal but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. To make the method robust with respect to high-contrast coefficients, we enrich the space solution via local eigenvalue problems, obtaining an optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes. The technique developed is dimensional independent and easy to extend to other elliptic problems such as elasticity.