spectralelement

Spectral element refers to a class of numerical methods for solving partial differential equations that combines the geometric flexibility of finite elements with the high accuracy of spectral methods. The computational domain is partitioned into nonoverlapping elements, and within each element the solution is approximated by high-order polynomials. The basis is typically composed of Lagrange polynomials associated with Gauss–Lobatto nodes, yielding a collocation or Galerkin formulation with favorable matrix structure. Solutions are usually enforced to be continuous across element interfaces, though discontinuous variants also exist.

In a spectral element method, each element uses a high-order polynomial basis, and the local mappings from a reference element enable curved geometries. Common element shapes are quadrilaterals in two dimensions and hexahedra in three dimensions, often implemented with tensor-product bases to exploit efficiency. The method supports hp-refinement, where increasing the polynomial degree (p) and refining the mesh (h) are combined to improve accuracy.

Convergence properties are a key feature: for smooth problems, spectral element methods achieve exponential (spectral) convergence with increasing polynomial degree, while for problems with reduced regularity the convergence is algebraic. The mass matrix can be diagonal or block-diagonal in collocation form, facilitating efficient time integration in explicit schemes, though implicit schemes are also used.

Advantages include high accuracy per degree of freedom, strong performance for wave-like and smooth solutions, and good parallel scalability. Limitations involve sensitivity to discontinuities and shocks, where additional stabilization or filtering is needed, as well as more complex implementation and mesh generation compared with low-order methods.

Applications span computational fluid dynamics, wave propagation, acoustics, electromagnetics, and elasticity. Variants include continuous spectral elements and discontinuous spectral elements, with hp-refinement as a common strategy.