High Order Finite Element MethodEdit

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High Order Finite Element Method

High Order Finite Element Method refers to a class of numerical techniques for approximating solutions to partial differential equations (PDEs) that employ high-degree polynomial basis functions on each element of a mesh. By using polynomials of degree p > 1, these methods aim to achieve higher accuracy per degree of freedom than traditional low-order approaches, especially for smooth solutions. The approach sits within the broader framework of the finite element method (FEM) and encompasses several variants and implementation strategies. See finite element method for the general context.

High order methods are widely used in engineering and the physical sciences because they can deliver exponential or algebraic rates of convergence for sufficiently smooth problems, with favorable performance when the solution has regular structure. They are not a universal replacement for low-order methods; their advantages depend on the problem regularity, the geometry, and the available computational resources. The development of high order methods has paralleled advances in mesh generation, numerical integration, and solver technology, and it interacts with related approaches such as isogeometric analysis and discontinuous Galerkin methods.

Overview

Concept: In high order FEM, each mesh element carries a basis of polynomials of degree p, rather than just linear or quadratic shapes. The global approximant is assembled by enforcing the PDE in a weak (variational) sense, typically using a Galerkin approach. See Galerkin method and weak form.
Key ideas: p-refinement (increasing the polynomial degree on fixed meshes), h-refinement (refining the mesh with lower-degree basis), and hp-adaptive strategies (combining both). See p-refinement, h-refinement, and hp-adaptive finite element method.
Variants: The most common categories are continuous Galerkin high order FEM, discontinuous Galerkin high order methods, and related spectral element formulations. See discontinuous Galerkin method and spectral element method.
Typical advantages: Higher accuracy per degree of freedom for smooth solutions, better dispersion properties in wave propagation problems, and the potential for fewer elements to achieve a target error compared with low-order methods. See discussions of convergence and error behavior in error estimation (numerical analysis).
Practical considerations: Increased cost per element due to larger local matrices and more complex quadrature, greater demand on mesh quality and element mapping, and potential conditioning challenges that motivate specialized solvers and preconditioners. See conditioning (numerical linear algebra) and Gauss quadrature.

Mathematical formulation

Weak form and function spaces: High order FEM is built on approximating the weak formulation of a PDE in appropriate function spaces, often Sobolev spaces such as H1 or similar, using polynomial basis functions on each element. See weak form and finite element method.
Reference element and mapping: The geometry is typically mapped from a reference element to each physical element via a smooth isomorphism, allowing the same local polynomial basis to be transported to the mesh. See reference element.
Basis functions: On each element, a set of high-degree polynomials is chosen, frequently built from nodal Lagrange points or hierarchical bases. Common choices include Legendre and Chebyshev polynomials in nodal form, with nodes chosen for stability and accuracy (e.g., Gauss or Gauss–Lobatto points). See Lagrange polynomials, Legendre polynomials, Chebyshev polynomials.
Discretization and assembly: The global system arises from the Galerkin projection of the PDE onto the finite element space, yielding a sparse linear or nonlinear system built from element stiffness matrices and load vectors. See stiffness matrix and Gauss quadrature for quadrature considerations.

Basis functions and discretization

p-refinement vs h-refinement: Increasing p on a fixed mesh (p-refinement) often yields rapid convergence for smooth solutions, while reducing element size (h-refinement) is more robust for problems with localized features. See p-refinement and h-refinement.
Nodal versus hierarchical bases: Nodal bases place degrees of freedom at specific points within each element, while hierarchical bases separate low- and high-order components to facilitate p-adaptivity and efficient solvers. See hierarchical basis (if available) and related discussions on basis choice.
Continuity and variants: Continuous Galerkin high order FEM enforces global continuity of the approximation, while discontinuous Galerkin (DG) methods allow discontinuities across element faces and use numerical fluxes to couple elements. Both families support high polynomial degree. See continuous Galerkin method and discontinuous Galerkin method.
Related formulations: Spectral element methods emphasize highly accurate polynomials with global-like behavior on each element, and isogeometric analysis (IGA) uses smooth basis functions from computer-aided design (CAD) representations, such as NURBS, in a FEM-like framework. See spectral element method and isogeometric analysis.

Variants and related methods

Spectral element method (SEM): Combines high-order polynomial approximation with a mesh of elements to achieve spectral accuracy for smooth problems. See spectral element method.
Isogeometric analysis (IGA): Integrates CAD-inspired basis functions with FEM, using splines or NURBS to achieve high continuity across elements. See isogeometric analysis.
Discontinuous Galerkin (DG): A high-order FEM variant where the solution is allowed to be discontinuous across element boundaries; numerical fluxes and stabilization terms enforce inter-element coupling. See discontinuous Galerkin method.
hp-adaptivity: Strategies that adaptively choose between refining the mesh and increasing the polynomial degree, guided by error indicators. See hp-adaptive finite element method.
Error estimation and adaptivity: A core aspect of high order FEM is a reliable error estimator to drive adaptive refinement and degree elevation. See error estimation (numerical analysis).

Numerical aspects

Numerical integration: Accurate quadrature is essential when using high-degree polynomials; the required quadrature order grows with p to integrate exactly the element matrices. See Gauss quadrature.
Conditioning and solver considerations: High-order discretizations can lead to larger and more ill-conditioned local matrices, prompting specialized preconditioners, block solvers, and matrix-free approaches. See conditioning (numerical linear algebra).
Mesh generation and geometry: Curvilinear elements and high-quality meshes help realize the promised accuracy gains; geometric approximation errors can dominate if mesh quality is poor. See mesh and mesh generation.
Computational cost and scalability: The per-element cost grows with p, but the reduction in the total number of elements can compensate under suitable conditions; parallel implementations emphasize data locality and communication efficiency. See general discussions of performance in numerical PDE literature and HPC resources.

Applications and domains

Structural mechanics and solid mechanics: High order elements are used to simulate stress, deformation, and vibration with high fidelity on complex geometries. See finite element method and isogeometric contexts.
Fluid dynamics and acoustics: For smooth flow fields and wave propagation problems, high order methods can reduce dispersion and diffusion errors, providing accurate results with fewer degrees of freedom in some regimes. See related topics in computational fluid dynamics.
Electromagnetics and wave propagation: In Maxwell’s equations and time-harmonic problems, high order methods can capture wave phenomena efficiently when the solution is smooth, with appropriate time integration and stabilization. See Maxwell's equations and wave equation.
Geophysics and seismic imaging: High order FEM supports accurate modeling of wave propagation through heterogeneous media, where hp-adaptivity can handle sharp interfaces and smooth regions alike. See seismic and wave propagation.

Controversies and debates (neutral overview)

When to use high order vs low order: The choice depends on smoothness of the solution, geometry complexity, and computational resources. For problems with strong discontinuities or sharp gradients, robust limiters, adaptive hp strategies, or switching to DG or lower-order methods may be preferred. See discussions in error estimation (numerical analysis) and discontinuous Galerkin method.
Conditioning and solver overhead: High order formulations can lead to larger, more ill-conditioned systems, raising the importance of preconditioning, solver design, and, in some cases, matrix-free techniques. See conditioning (numerical linear algebra).
Geometry representation: While high order methods benefit from accurate geometry representation, integrating CAD-based geometries via IGA can raise implementation complexity and licensing considerations; the trade-offs are an active area of research. See isogeometric analysis.
Applicability to non-smooth problems: For problems with shocks, singularities, or highly irregular solutions, high order methods may require limiters or hybrid approaches; in some cases, lower-order methods with adaptive refinement can be more robust. See hp-adaptive finite element method and discontinuous Galerkin method discussions.