# Numerical homogenization of elliptic PDEs with similar coefficients

## TL;DR

This paper introduces a parallelizable Petrov-Galerkin localized orthogonal decomposition algorithm for efficiently solving sequences of elliptic PDEs with similar, rapidly varying coefficients, applicable in time-dependent and stochastic contexts.

## Contribution

The paper develops an adaptive PG-LOD method that selectively recomputes local correctors, improving efficiency for sequences of similar elliptic PDEs.

## Key findings

- The method effectively handles sequences with similar coefficients.
- Adaptive recomputation enhances computational efficiency.
- Application demonstrated on 3D time-dependent Darcy flow.

## Abstract

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.