# Numerical upscaling of perturbed diffusion problems

## TL;DR

This paper introduces a numerical method for efficiently solving perturbed elliptic PDEs with rapidly varying coefficients by reusing local computations, leveraging PG-LOD for parallelization and low memory use.

## Contribution

The paper develops a PG-LOD based approach for perturbed diffusion problems, enabling efficient reuse of computations for local defects and global mappings.

## Key findings

- Method achieves efficient solutions for perturbed problems.
- Parallelization reduces computational overhead.
- Numerical examples demonstrate accuracy and efficiency.

## Abstract

In this paper we study elliptic partial differential equations with rapidly varying diffusion coefficient that can be represented as a perturbation of a reference coefficient. We develop a numerical method for efficiently solving multiple perturbed problems by reusing local computations performed with the reference coefficient. The proposed method is based on the Petrov--Galerkin Localized Orthogonal Decomposition (PG-LOD) which allows for straightforward parallelization with low communcation overhead and memory consumption. We focus on two types of perturbations: local defects which we treat by recomputation of multiscale shape functions and global mappings of a reference coefficient for which we apply the domain mapping method. We analyze the proposed method for these problem classes and present several numerical examples.