p-FEM or the p-version of the

finite element method Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...

is a

numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...

for solving

partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...

. It is a discretization strategy in which the finite element mesh is fixed and the polynomial degrees of elements are increased such that the lowest polynomial degree, denoted by

p_

, approaches infinity. This is in contrast with the "h-version" or "h-FEM", a widely used discretization strategy, in which the polynomial degrees of elements are fixed and the mesh is refined such that the diameter of the largest element, denoted by

h_

approaches zero. It was demonstrated on the basis of a linear elastic fracture mechanics problem that sequences of finite element solutions based on the p-version converge faster than sequences based on the h-version by

Szabó Szabó () is a common Hungarian language, Hungarian surname, meaning "tailor". In 2019, it occurred in 203,126 names, making it the fourth most frequent Hungarian names, Hungarian surname. In Czech language, Czech and Slovak language, Slovak, a fem ...

and

Mehta Mehta () is an Indian surname, derived from the Sanskrit word ''mahita'' meaning 'great' or 'praised'. It is found among several Indian religious groups, including Hindus, Jains, Parsis, and Sikhs. Among Hindus, it is used by a wide range of castes ...

in 1978. The theoretical foundations of the p-version were established in a paper published Babuška, Szabó and Katz in 1981 where it was shown that for a large class of problems the asymptotic rate of convergence of the p-version in energy norm is at least twice that of the h-version, assuming that quasi-uniform meshes are used. Additional computational results and evidence of faster convergence of the p-version were presented by Babuška and Szabó in 1982. The distinction between the h- and p-versions exists primarily for historical and theoretical reasons. In practical applications the design of the mesh and the choice polynomial degrees are both important. In fact, it is possible to realize exponential rates of convergence when the p-version is used in combination with proper mesh design. This point was discussed from the engineering perspective by Szabó and from the theoretical perspective by Guo and Babuška in 1986. Realization of exponential rates of convergence for Maxwell equations was discussed by Costabel, Dauge and Schwab in 2005Costabel, M., Dauge, M., and Schwab, C., "Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains." Mathematical Models and Methods in Applied Sciences 15(04), pp. 575-622, 2005.

References

{{reflist Finite element method Numerical differential equations Partial differential equations