Local Finite Elements

Updated 2 July 2025

Local Finite Elements are methods using basis functions with compact support to enable efficient and adaptive computation in spatially localized regions.
They leverage localized error estimation and adaptive refinement to robustly tackle multiscale, high-contrast, and interface problems in numerical PDEs.
By decomposing global challenges into independent local computations, these methods enhance parallelization, solver performance, and overall computational efficiency.

Local finite elements refer to finite element methods and constructions in which basis functions, approximation strategies, or assembly processes are designed to be spatially localized—either through explicit compact support, exploitation of locality in mesh topology, or decomposition of large or multiscale problems into local computations. In modern theory and applications, this concept has expanded to encompass localized error control, adaptivity, and advanced algebraic and geometric formulations, including their interplay with mesh structures, PDE regularity, and computational strategies.

1. Locality in Finite Element Spaces and Approximations

Local finite elements are characterized by basis functions or approximation operators that are nonzero only in restricted, localized regions of the computational mesh. This property facilitates efficient computation, enables adaptive and parallelizable algorithms, and plays a central role in theoretical error analysis and mesh design.

Support of Basis Functions: In classical finite element methods, basis functions associated with mesh vertices or edges naturally exhibit compact support. Local finite elements generalize this idea by constructing basis functions whose nonzero support is deliberately aligned with geometric or physical structures (e.g., layers, interfaces, inclusions, or local mesh patches).
Mesh Patch Locality: Localized multiscale and adaptive methods define node patches, element pairs, or oversampled neighborhoods as the domain of support for local computations, basis construction, or error estimation (1703.06325, 2211.09461). The adaptively constructed AL basis, for example, produces problem-adapted, locally supported functions for each coarse mesh node.
Element Splitting: When interfaces or discontinuities are present, local refinement is performed within affected elements, keeping the global mesh structure unchanged and modifying only the minimal required neighborhood—see, e.g., locally adapted patch methods (1610.00023) and locally modified fitted methods (1806.00999, 2007.13906).

2. Localized Basis Construction and Multiscale Methods

A central development within this field is the systematic derivation of multiscale, problem-adapted local basis functions to address PDEs with high heterogeneity or multiple scales.

Adaptive Local (AL) Basis and Generalized Local Snapshots: In the AL basis methodology (1703.06325), problem-adapted basis functions are constructed by solving (discrete) local problems on patches around each mesh node, splitting the problem into "nearfield" (local roughness) and "farfield" (decayed, harmonic) components. The number of local basis functions per node scales like $O(\log(1/H)^{d+1})$ , supporting optimal convergence for rough coefficients.
Super-Localized Generalized Finite Element Methods (SGFEM): These methods (2211.09461) employ a partition of unity framework and local snapshot spaces, constructing basis functions by solving local Dirichlet problems on oversampled patches and then projecting onto an optimal reduced subspace via a local eigenproblem (Kolmogorov $n$ -width). The use of partition of unity ensures basis stability and seamless global assembly.
Multiscale Mixed FEM and LOD: In mixed finite element problems, the localized orthogonal decomposition (LOD) technique (1501.05526) enriches coarse spaces with exponentially decaying local correctors, constructed through independent local Raviart-Thomas (RT) or Nédélec problems in small patches. This localization enables efficient, parallelizable solution of heterogeneous multiscale PDEs.

3. Local Error Estimation and Robust Adaptivity

Robust, parameter- or structure-independent a priori and a posteriori error estimates are foundational for adaptive mesh refinement.

Localization of the Best Error: For reaction-diffusion problems, the global best approximation error is robustly equivalent to a sum of local best errors computed over pairs of elements sharing an interface, rather than on single elements (1402.3959). This pair-wise localization strategy remains robust as diffusion or reaction coefficients vary and is essential for designing effective local error functionals in adaptive tree algorithms.
Broken Bramble-Hilbert Lemma and Local Interpolants: The development of local versions of the Bramble-Hilbert lemma, as for finite element differential forms (2011.00634), enables precise local error estimates in terms of mesh size and local Sobolev regularity. Generalized Clément and Scott-Zhang interpolants, adapted for vector-valued FE spaces, maintain minimal smoothness assumptions and boundary compatibility.

4. Algebraic, Geometric, and Functional Perspectives

Recent research has deepened the algebraic and geometric underpinnings of local finite element constructions, especially in vector-valued and exterior calculus contexts.

Finite Element Exterior Calculus (FEEC): The FEEC framework treats finite element spaces as subcomplexes of spaces of differential forms, preserving geometric and algebraic properties of PDEs. Local finite element approximation is underpinned by bounded commuting projections, local error estimates, and the construction of higher-order and non-uniform polynomial order de Rham complexes (2310.10479, 2011.00634).
Partially Localized Flux Reconstructions: In equilibrated a posteriori estimation and flux recovery, the problem is split into a global lowest-order part and a collection of independent local higher-order corrections. This approach is both theoretically robust (preserving cohomological properties) and computationally efficient by relegating most of the heavy computation to local simplex-based problems.
Nonconforming Local Elements: Families of nonconforming elements (e.g., Crouzeix-Raviart type in 3D (1703.03224)) are constructed via explicit, local polynomials supported on one or two adjacent simplices and are characterized by carefully designed jump conditions across faces. Orthogonal polynomials and symmetry decompositions play a key role in ensuring optimal approximation and local support.

5. Locality for Interface, Polygonal, and Anisotropic Mesh Methods

Strategies for handling interfaces, corners, anisotropy, and complex mesh geometries rely heavily on local finite element constructions.

Locally Modified, Fitted, and Patch-based Approaches: Interface-fitted and locally modified finite element methods (1806.00999, 2007.13906, 1610.00023) apply local mesh subdivision and parametric geometry alterations solely within patches intersected by interfaces, retaining optimal approximation properties and conditioning. Isoparametric mappings are used to accommodate curved interfaces.
Nyström-based and Trefftz Local Spaces: On polygonal or curvilinear elements, local finite element spaces are defined as the solution set of Poisson or Laplace problems with polynomial or non-polynomial boundary data (1708.07323, 1906.09015). Nyström discretization of boundary integral equations enables high-accuracy local solution, even for singular or complex base geometries. The resulting basis ensures optimal convergence without global mesh refinement or adaptivity.
Anisotropic Local Fitting and Mixed Meshes: For interface problems with anisotropy, local mesh adaptation yields elements of high aspect ratio only in the near-interface region (2005.05774). Finite element spaces are defined accordingly (e.g., with both triangles and quadrilaterals) and optimal interpolation and solver properties are maintained through rigorous mesh quality criteria and multigrid algorithms.

6. Connections to Algebraic Structures, Finite Volumes, and Efficiency

Connections between finite element and finite volume schemes are often established through local algebraic constructions.

Petrov-Galerkin RT and Local FV Equivalence: By constructing a local dual basis to the Raviart-Thomas space, the Petrov-Galerkin mixed FE method realizes a discrete gradient operator whose computation is strictly local—depending only on adjacent cells—thus making the method algebraically and computationally equivalent to a locally conservative finite volume method such as VF4 (1712.08006).
Tensor Product and Symbolic Manipulation: Tensor product finite elements, constructed via local algebraic and UFL-based symbolic structures, enable efficient element assembly, mixed/complex element spaces, and local code generation for product cell types (e.g., quadrilaterals, prism, hexahedra) in computational frameworks (1411.2940).

7. Numerical and Practical Impact

Local finite elements underpin a broad range of computational advances and practical algorithms:

Efficiency: Locality enables parallelization, memory efficiency, and reduced communication, especially when local correctors or basis functions are computed independently (1501.05526, 2211.09461).
Mesh Flexibility: The techniques enable the handling of complex, evolving, or partially damaged meshes (1808.06350), as well as the robust extension to higher-order elements, hp-adaptivity, and intricate geometric domains.
Solver Performance: Approaches that focus on locality, basis scaling, or hierarchical representations are crucial for controlling the conditioning of system matrices and thereby the performance of iterative solvers (1806.00999, 1703.06325).

Aspect	Local Finite Element Approach/Benefit
Basis/locality	Compact support, adaptive patch-based functions, mesh partitioning
Error estimates	Localized best error, broken Bramble-Hilbert, patchwise interpolants
Applications	Multiscale elliptic, interface, high-contrast, electromagnetic, and singular problems
Algorithmic/implementation property	Parallelizability, patchwise independence, adaptability, memory efficiency
Geometric/algebraic frameworks	FEEC, tensor product structure, nonconforming local bases, flux reconstructions

Local finite elements constitute a foundational and versatile set of methods, constructions, and theoretical tools in modern numerical PDE and scientific computing. By focusing on the localization of basis functions, error estimation, and algebraic structure, researchers have enabled efficient, scalable, and robust solution strategies that address the most challenging classes of multiscale, interface, and high-heterogeneity problems across science and engineering.