Hp-FEM

The symbol hp-FEM is an abbreviation for a general version of the
Finite Element Method that employs elements of variable size "(h)" and polynomial degree "(p)". The origins of this method date back to the pioneering work of Ivo Babuska et. al who discovered that the finite element method converges "exponentially fast" whenthe 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 methodswhich only converge with an algebraic rate. The exponential convergenceof the hp-FEM was not only predicted theoretically but also observed by numerous independent researchers.

Differences from standard FEM

The hp-FEM differs from the standard (lowest-order) FEM in many aspects.
* Choice of higher-order shape functions: To begin with, the higher-degree polynomials in elements can be generated using different sets of shape functions. The choice of such set can influence dramatically the conditioning of the stiffness matrix, and in turn the entire solution process.
* Automatic hp-adaptivity: In the hp-FEM, an element can be hp-refined in many different ways. One way is to just increase its polynomial degree without subdividing it in space. Or, the element can be subdivided geometrically, and various polynomial degrees can be applied to the subelements. The number of element refinement candidates easily reaches 100 in 2D and 1000 in 3D. Therefore, clearly, one number indicating the size of error in an element is not enough to guide automatic hp-adaptivity (as opposed to adaptivity in standard FEM). Other techniques such as "reference solutions" or "analyticity considerations" must be employed to obtain more information about the "shape of error" in every element.
* Ratio of assembling and solution CPU times: In standard FEM, the stiffness matrix usually is assembled quickly but it is quite large. Therefore, typically, the solution of the discrete problem consumes the largest part of the overall computing time. On the contrary, the stiffness matrices in the hp-FEM typically are much smaller, but (for the same matrix size) their assembly takes more time than in standard FEM. Mostly, this is due to the computational cost of higher-order numerical quadrature.
* Analytical challenges: The hp-FEM is more difficult to understand from the analytical point of view than standard FEM. This concerns numerous techniques, such as the discrete maximum principles (DMP) for elliptic problems. These results state that, usually with some limiting assumptions on the mesh, the piecewise-polynomial FEM approximation obeys analogous maximum principles as the underlying elliptic PDE. Such results are very important since they guarantee that the approximation remain physically admissible - for example, that we do not compute a negative density, negative concentration, or negative absolute temperature. The DMP are quire well understood for lowest-order FEM but completely unknown for the hp-FEM in two or more dimensions. First DMP in one spatial dimension were formulated recently by Solin and Vejchodsky.
* Programming challenges: It is much harder to implement a hp-FEM solver than standard FEM code. The multiple issues that need to be overcome include (but are not limited to): higher-order quadrature formulas, higher-order shape functions, connectivity and orientation information relating shape functions on the reference domain with basis functions in the physical domain, etc.

Higher-order shape functions

In standard FEM one only works with shape functions associated with grid vertices (the so-called "vertex functions"). In contrast to that,in the hp-FEM one employs "vertex functions", "edge functions" (associated with element edges), "face functions" (corresponding to element faces - 3D only),and "bubble functions" (higher-order polynomials which vanish on element boundaries).

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Example: the Fichera Problem

TODO

Open Source hp-FEM Codes

* [http://www.concepts.math.ethz.ch/ Concepts] : C/C++ hp-FEM/DGFEM/BEM library for elliptic equations developed at ETH Zurich (Switzerland)
* [http://users.ices.utexas.edu/~leszek/2dhp90.html 2dhp90] , [http://users.ices.utexas.edu/~leszek/3Dhp90.html 3dhp90] : Fortran codes for elliptic problems and Maxwell's equations developed by L. Demkowicz at ICES, UT Austin.
* [http://spilka.math.unr.edu/projects/hermes2d Hermes] : C/C++ library for rapid prototyping of space- and space-time adaptive hp-FEM solvers, with application to a wide range of PDEs and multiphysics PDE systems. Developed by the [http://spilka.math.unr.edu/ hp-FEM group] at the University of Nevada, Reno (USA) and Institute of Thermomechanics in Prague (Czech Republic).

References

* I. Babuska, B.Q. Guo: The h, p and h-p version of the finite element method: basis theory and applications, Advances in Engineering Software, Volume 15, Issue 3-4, 1992.
* I. Babuska, M. Suri: The p- and h-p versions of the finite element method, basic principles and properties. SIAM Review, Volume 36, Issue 4, 1994
* I. Babuska, M. Suri: The p- and h-p version of the finite element method, an overview, Computer Methods in Applied Mechanics and Engineering, Volume 80, Issue 1-3, 1990.
* W. Gui, I. Babuska: The h, p and h-p versions of the finite element method in 1 dimension. Part 1. The error analysis of the p-version. Numerische Mathematik, Volume 49, Issue 6, 1986.
* W. Gui, I. Babuska: The h, p and h-p versions of the finite element methods in 1 dimension . Part III. The adaptive h-p version.Numerische Mathematik, Volume 49, Issue 6, 1986.
* L. Demkowicz, J. Kurtz, D. Pardo, W. Rachowicz, M. Paszenski, A. Zdunek: Computing with hp-Adaptive Finite Elements, Chapman & Hall/CRC Press, 2007.
* J.M. Melenk: hp-Finite Element Methods for Singular Perturbations, Springer, 2002.
* C. Schwab: p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Oxford University Press, 1998.
* P. Solin: Partial Differential Equations and the Finite Element Method, J. Wiley & Sons, 2005.
* P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, 2003.
* P. Solin, T. Vejchodsky: A Weak Discrete Maximum Principle for hp-FEM, J. Comput. Appl. Math. 209 (2007) 54-65.
* T. Vejchodsky, P. Solin: Discrete Maximum Principle for Higher-Order Finite Elements in 1D, Math. Comput. 76 (2007), 1833 - 1846.