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{{short description|Numerical method for solving partial differential equations}}
{{lowercase}}
'''p-FEM''' or the p-version of the [[finite element method]] is a [[Numerical analysis|numerical method]] for solving [[partial differential equations]]. 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 <math>p_{\min}</math>, 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 <math>h_{\max}</math> 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 [[Barna Szabó|Szabó]] and [[Mehta A. K.|Mehta]] in 1978.<ref>Szabó, B. A. and Mehta, A. K., "p-Convergent Finite Element Approximations in Fracture Mechanics." International Journal for Numerical Methods in Engineering 12, pp. 551-560, 1978.</ref> The theoretical foundations of the p-version were established in a paper published [[Ivo Babuska|Babuška]], Szabó and [[Katz, I. N.|Katz]] in 1981<ref>Babuška I, Szabó, B. A. and Katz, I. N., "The p-version of the finite element method." SIAM Journal on Numerical Analysis 18, pp. 515-545, 1981.</ref> 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.<ref>Babuška, I. and Szabó, B. A., "On the Rates of Convergence of the Finite Element Method." International Journal for Numerical Methods in Engineering 18, pp. 323-341, 1982.</ref>

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, B|Guo]] and Babuška in 1986.<ref>Szabó, B. A., "Mesh design for the p-version of the finite element method." Computer Methods in Applied Mechanics and Engineering 55(1), pp. 181-197, 1986.</ref><ref>Guo, B. and Babuška, I. "The h-p version of the finite element method. Part 1. Basic approximation results." Computational Mechanics 55, pp. 21-41, 1986.</ref> Realization of exponential rates of convergence for Maxwell equations was discussed by [[Costabel, M.|Costabel]], [[Dauge, M.|Dauge]] and [[Schwab, C.|Schwab]] in 2005<ref>Costabel, 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.</ref>

==References==
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[[Category:Finite element method]]
[[Category:Numerical differential equations]]
[[Category:Partial differential equations]]