Adaptive Mesh Refinement for Spectral Element problems

Picture of rhys-hawkins.md Rhys Hawkins

7 Dec 2023

This project is intended as a full year Honours project and a scholarship is available to the right candidate.

The spectral element method (SEM) is a widely used variant of finite elements adapted to quadrilateral (2D) or hexahedral (3D) elements. It’s advantages are that it is relatively simple to implement, the use of tensorized bases in higher dimensions is computationally efficient including higher order elements, and it produces a diagonal mass matrix which is useful for explicit solutions of time varying problems.

The down side of the use of quads or hex elements is that meshes are difficult to construct automatically. In addition, when there is a sharp change in material properties, e.g. ground - atmosphere, traditional spectral element meshes are highly inefficient.

There have been recent attempts at incorporating Adaptive Mesh Refinement (AMR) into standard spectral element codes. The core of technique required by these methods are non-conforming spectral elements, where the edges/faces of neighboring elements do not precisely align. Non-conforming meshes allow greater flexibility and simplicity in mesh generation, but a large problem is that the mass matrix will no longer be diagonal, rather it will be block diagonal and therefore more difficult to invert.

This project will examine accuracy and performance issues of an implementation of spectral elements with non-conforming cells in time varying problems at different scales.

The target application is in geophysical modelling where we often have sharp changes in material properties, for example, ground-atmosphere or mantle-outer core.

More information is available on my homepage

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