The p-version of the finite element method for problems requiring \(C^ 1\)-continuity. (English) Zbl 0602.65086

Katz, I. Norman; Wang, Douglas W.

SIAM J. Numer. Anal. 22, 1082-1106 (1985).

The p-version of the finite element method consists on increasing to infinity the polynomial degree p of the spline approximants while keeping the subdivision of the domain fixed.
From authors’ introduction: We formulate the mathematical model of the p- version of the finite element method for the case of \(C^ 1\)-continuity. We choose the biharmonic equation as a model problem, and we then analyze the convergence rate of the p-version. By using previously derived results for functional approximation, we are able to modify the approximation in such a way that it satisfies the boundary conditions and continuity requirements in each subdomain. We then investigate a priori estimates for these approximate solutions. The convergence rates for smooth solutions and for solutions which have a singularity caused by angular boundary conditions are presented. The techniques used are similar to those in [I. Babushka, B. A. Szabo and the first author [ibid. 18, 515-545 (1981; Zbl 0487.65059)] but significant changes are required for these techniques to be successful in the \(C^ 1\)-case. Finally we report the results of testing our program on some benchmark problems in order to examine its performance. Sample results are presented and compared with theoretical predictions.

Reviewer: H.Marcinkowska

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
31A30	Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40	Boundary value problems for higher-order elliptic equations

Keywords:

finite element method; biharmonic equation; convergence rate; smooth solutions; singularity; benchmark problems

Citations:

Zbl 0487.65059