# A domain mapping approach for elliptic equations posed on random bulk and surface domains

## TL;DR

This paper develops a domain mapping method to efficiently approximate statistical moments of solutions to elliptic PDEs on random geometries, including surfaces and bulk-surface systems, with proven error estimates and numerical validation.

## Contribution

It introduces a geometric analysis framework for reformulating elliptic equations on random surfaces onto a fixed domain, enabling finite element and Monte Carlo methods with optimal error bounds.

## Key findings

- Theoretical convergence rates are established.
- Numerical experiments confirm the accuracy of the method.
- Applicable to both surface and bulk-surface elliptic problems.

## Abstract

In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fix deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.