Hp-FEM

hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size (h) and polynomial degree (p). The origins of hp-FEM date back to the pioneering work of Ivo Babuska et al. who discovered that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and p-refinements (increasing their polynomial degree). The exponential convergence makes the method a very attractive choice compared to most other finite element methods which only converge with an algebraic rate. The exponential convergence of the hp-FEM was not only predicted theoretically but also observed by numerous independent researchers.

The hp-FEM differs from the standard (lowest-order) FEM in many aspects.

The Fichera problem (also called the Fichera corner problem) is a standard benchmark problem for adaptive FEM codes. One can use it to show the dramatic difference in the performance of standard FEM and the hp-FEM. The problem geometry is a cube with missing corner. The exact solution has a singular gradient (an analogy of infinite stress) at the center. The knowledge of the exact solution makes it possible to calculate the approximation error exactly and thus compare various numerical methods. For illustration, the problem was solved using three different versions of adaptive FEM: with linear elements, quadratic elements, and the hp-FEM.

Fichera problem: singular gradient.

Fichera problem: convergence comparison.

The convergence graphs show the approximation error as a function of the number of degrees of freedom (DOF). By DOF we mean (unknown) parameters that are needed to define the approximation. The number of DOF equals the size of the stiffness matrix. The reader can see in the graphs that the convergence of the hp-FEM is much faster than the convergence of both other methods. Actually, the performance gap is so huge that the linear FEM might not converge at all in reasonable time and the quadratic FEM would need hundreds of thousands or perhaps millions of DOF to reach the accuracy that the hp-FEM attained with approximately 17,000 DOF. Obtaining very accurate results using relatively few DOF is the main strength of the hp-FEM.

Smooth functions can be approximated much more efficiently using large high-order elements than small piecewise-linear ones. This is illustrated in the figure below, where a 1D Poisson equation with zero Dirichlet boundary conditions is solved on two different meshes. The exact solution is the sin function.

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