# Approximation of Periodic PDE Solutions with Anisotropic Translation Invariant Spaces

## TL;DR

This paper presents a unified framework for approximating solutions to periodic PDEs in linear elasticity using anisotropic translation invariant spaces, extending FFT-based methods and connecting them to finite element approaches.

## Contribution

It introduces a general approach that unifies various FFT-based discretizations and extends them to anisotropic lattices, linking to finite element methods.

## Key findings

- Demonstrates the numerical advantages of the proposed framework.
- Shows the connection between discrete solution spaces and finite element methods.
- Extends FFT-based discretizations to anisotropic lattices.

## Abstract

We approximate the quasi-static equation of linear elasticity in translation invariant spaces on the torus. This unifies different FFT-based discretisation methods into a common framework and extends them to anisotropic lattices. We analyse the connection between the discrete solution spaces and demonstrate the numerical benefits. Finite element methods arise as a special case of periodised Box spline translates.