# Extrapolation in the finite element method with penalty

• J. Thomas King
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 430)

## Abstract

Consider the model problem Δu=f in Ω, u=0 on δΩ. Here Ω is a bounded open subset of Rn with smooth boundary, δΩ. The penalty method provides a method for obtaining an approximate solution without requiring the approximant to satisfy boundary conditions. Unfortunately, we pay a price for this convenience, namely loss of accuracy. We show that this difficulty may be alleviated by a particular type of extrapolation process. For a particular choice of boundary weight in the penalty method we show that repeated extrapolation always yields "optimal" error estimates in the energy norm.

## Keywords

Finite Element Method Penalty Method Essential Boundary Condition Satisfy Boundary Condition Optimal Error Estimate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Bibliography

1. 
J. P. Aubin, Approximation des problèms aux limites non homogenes et régularité de la convergence, Calcolo, 6(1969), pp. 117–139.
2. 
I. Babuška, The finite element method with penalty, Math. Comp., 27(1973), pp. 221–228.
3. 
_____, Approximations by hill functions, Comment. Math. Univ. Carolinae, 11(1970), pp. 787–811.
4. 
A. Berger, R. Scott, G. Strang, Approximate boundary conditions in the finite element method, Symposia Mathematica, Academic Press, New York, 1972, pp. 295–313.Google Scholar
5. 
J. H. Bramble and S. Hilbert, Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation, SIAM Num. Anal., 7(1970), pp. 112–124.
6. 
J. H. Bramble and J. Nitsche, A generalized Ritz-least-squares method for Dirichlet problems, SIAM Num. Anal., 10(1973), pp. 81–93.
7. 
J. H. Bramble and A. H. Schatz, Releigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions, Comm. Pure Appl. Math., 23(1970), pp. 653–675.
8. 
__________, Least-squares methods for 2mth order elliptic boundary value problems, Math. Comp., 25(1970), pp. 1–33.
9. 
J. H. Bramble, M. Zlamal, Triangular elements in the finite element method, Math. Comp., 24(1970), pp. 809–820.
10. 
R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49(1942), pp. 1–23.
11. 
J. E. Dendy, Penalty Galerkin methods for partial differential equations, Ph.D. thesis, Rice University, 1971.Google Scholar
12. 
G. Fix, K. Larsen, On the convergence of SOR iterations for finite element approximations to elliptic boundary value problems, SIAM Num. Anal., 8(1971), pp. 536–547.
13. 
J. T. King, New error bounds for the penalty method and extrapolation, Numer. Math., to appear.Google Scholar
14. 
J. L. Lions, E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968.
15. 
J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press (A. K. Aziz, editor), New York, 1972, pp. 603–627.
16. 
M. H. Schultz, Multivariate spline functions and elliptic problems, SIAM Num. Anal., 6(1969), pp. 523–538.
17. 
S. Serbin, A computational investigation of least squares and other projection methods for the approximate solution of boundary value problems, Ph.D. thesis, Cornell University, 1971.Google Scholar