The Spectral Element Method
A hybrid that combines the geometric flexibility of finite elements with the exponential accuracy of spectral methods.
Best of both worlds
The spectral element method divides the domain into elements like the finite element method, but within each element it represents the solution with high-order polynomials as in spectral methods. This gives it the ability to mesh complex geometries while retaining, within each element, the exponential convergence of spectral discretization for smooth solutions. It is a leading choice for high-fidelity simulation of smooth flows.
High-order nodes within elements
Each element carries a set of nodes placed at Gauss-Lobatto-Legendre points, at which high-degree Lagrange polynomials are defined. These node placements avoid the Runge oscillation that plagues equally spaced high-order interpolation, and they coincide with a Gauss quadrature rule, so integration and interpolation share the same points. This coincidence produces a diagonal mass matrix, which makes explicit time stepping efficient.
- h-refinement: add more elements to resolve geometry or features
- p-refinement: raise the polynomial degree within elements for smooth regions
- Diagonal mass matrix from Gauss-Lobatto nodes speeds explicit time integration
- Exponential convergence per element when the solution is smooth
Convergence and conditioning
For smooth solutions, refining by raising polynomial degree (p-refinement) gives exponential convergence, while refining by adding elements (h-refinement) gives algebraic convergence but handles geometry and non-smoothness. Combining them (hp-adaptivity) is optimal. The trade-off is that very high element order raises the operator's condition number and demands careful preconditioning of the resulting systems.
Use in simulation
The spectral element method dominates high-fidelity incompressible and low-Mach flow simulation and direct numerical simulation of turbulence, where its low numerical dissipation preserves fine-scale physics. It is used in plasma and fusion contexts for accurate wave propagation and flow modeling, wherever smooth fields on nontrivial geometry demand both accuracy and flexibility. Discontinuous Galerkin methods are a closely related high-order element family that relax inter-element continuity to better handle discontinuities.