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

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.

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 |