Hprefinement

hprefinement, usually written hp-refinement in the literature, is an adaptive discretization strategy used in finite element analysis to improve numerical accuracy by jointly refining the mesh (reducing element size h) and increasing the local polynomial degree p of the basis functions. The method is designed to handle diverse solution regularities by choosing h- or p-refinement locally according to an a posteriori error estimator or indicator. In smooth regions, increasing p (p-refinement) can yield exponential convergence, while near singularities or steep gradients, reducing h (h-refinement) may be more effective. The combination, hp-refinement, allows for flexible adaptation and can achieve high accuracy with relatively few degrees of freedom compared with uniform refinement.

In practice, hp-refinement proceeds iteratively: solve on the current discretization, estimate the error, select elements for refinement and whether to refine h or p, perform the refinements, and repeat until the target error is met. Error estimators may be residual-based, goal-oriented, or based on normalized fluxes. Implementations exist in many finite element packages and are central to hp-FEM and spectral element methods. The approach requires careful handling of stability and conditioning for high-order polynomials and to avoid ill-conditioned stiffness matrices. Benefits include fast convergence for smooth problems and robust handling of complex geometries or localized features. Limitations include increased algorithmic complexity, higher assembly costs for high-order elements, and the need for sophisticated mesh-generation tools.

Historically, the hp-FEM framework was developed in the 1990s by researchers including Ivo Babuska and Ulrich Melenk, and has since become a standard technique for adaptive high-order discretizations. See also: hp-FEM, adaptive mesh refinement, h-refinement, p-refinement, finite element method.