Numerical homogenization of elliptic PDEs with similar coefficients
For researchers solving sequences of elliptic PDEs with similar coefficients (e.g., in time-dependent or stochastic problems), this algorithm reduces computational cost by avoiding full recomputation.
The paper proposes a parallelizable algorithm for numerically homogenizing elliptic PDEs with similar rapidly varying coefficients, achieving adaptive recomputation of local corrector problems only where needed. The method is demonstrated on a 3D time-dependent two-phase Darcy flow problem.
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per time step) and in sample based stochastic integration of outputs from an elliptic PDE (with one coefficient per sample member). We propose a parallelizable algorithm based on Petrov-Galerkin localized orthogonal decomposition (PG-LOD) that adaptively (using computable and theoretically derived error indicators) recomputes the local corrector problems only where it improves accuracy. The method is illustrated in detail by an example of a time-dependent two-pase Darcy flow problem in three dimensions.