The Spectral Element Method
Spectral elements combine finite-element geometric flexibility with high-order polynomial accuracy inside each element.
High order meets unstructured meshes
The spectral element method (SEM) partitions the domain into elements like the finite element method, but inside each element it uses high-degree polynomial basis functions placed at carefully chosen nodes, typically Gauss-Lobatto-Legendre points. This delivers spectral-like convergence on smooth solutions while retaining the ability to mesh complex geometry element by element.
The Gauss-Lobatto nodes serve double duty: they define the interpolating basis and the quadrature rule. This collocation choice makes the elemental mass matrix diagonal, which greatly simplifies explicit time stepping.
Convergence knobs
SEM offers two convergence paths. Refining the mesh (more, smaller elements) gives algebraic convergence controlled by element size; raising the polynomial degree within elements gives exponential convergence when the solution is smooth. Practitioners tune both, using low order near singularities and high order where the solution is smooth.
Accuracy and dispersion
High-order elements have very low numerical dispersion and dissipation, so waves propagate accurately over long distances and many periods with little phase error. This makes SEM attractive for wave propagation, incompressible flow, and turbulence where preserving spectral content over long integrations matters.
- Gauss-Lobatto-Legendre nodes for basis and quadrature
- Diagonal mass matrix eases explicit time integration
- Low dispersion error for long-time wave propagation
- Continuous (CG) and discontinuous (DG) elemental couplings
SEM sits between global spectral methods and standard finite elements, capturing much of the accuracy of the former with the geometric flexibility of the latter. It is well suited to fluid and MHD problems in shaped fusion-relevant domains where both accuracy and geometry matter.