# A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains

## TL;DR

This paper develops a comprehensive theoretical framework for finite element methods applied to partial differential equations on evolving domains, including surfaces and bulk systems, with proven optimal bounds and numerical validation.

## Contribution

It introduces a unified abstract variational theory for continuous in time finite element discretizations on evolving domains, including curved spaces and coupled surface-bulk systems.

## Key findings

- Proved optimal a priori error bounds for the discretizations.
- Developed evolving finite element spaces on flat and curved geometries.
- Numerical experiments confirm theoretical convergence rates.

## Abstract

We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on evolving surfaces and bulk domains as well coupled surface bulk systems. We use an abstract variational setting with time dependent function spaces and abstract time dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described which confirm the rates of convergence.