THE H-VERSION, P-VERSION AND HP-VERSION OF THE FINITE-ELEMENT METHOD

The finite element method developed by engineers in 1950 was considered in the first three decades as an h-version, that means the degree p of the polynomials used was fixed at a low rate (e.g. p = 1, 2) and the mesh size h was reduced to obtain a better approximation of the exact solution. In 1981, I. BABUSKA, B. A. SZABO and I. N. KATZ introduced the p-version: they fixed the mesh size h and increased the degree p of the polynomials in order to reduce the approximation error. In the same year, I. BABUSKA and M. R. DORR investigated a combination of both versions and tried it the hp-version. Here both the mesh size h and the polynomial degree p are changed to improve accuracy. This was done with the help of an error estimator needed to decide where the degree p should be increased (p --> p + 1) or which finite element must be refined (h --> HBAR < h). The three versions lead to different approximation results. While the h-version and the p-version have only a polynomial rate of convergence with respect to the number of degrees of freedom, the hp-version with a geometric mesh refinement has an exponential rate.