The h-p version of the finite element method with quasiuniform meshes. (English) Zbl 0623.65113

Babuška, I.; Suri, Manil

RAIRO, Modélisation Math. Anal. Numér. 21, 199-238 (1987).

h-p version denotes achieving accuracy by both refining the mesh (decreasing h) and increasing the degree p of the elements. Error estimates of the type \[ \| u-u_{hp}\|_ 1\leq C(h^{\mu - 1}/p^{k-1})\| u\|_ k,\quad \mu =\min (k,p+1) \] are obtained for the model problem \(-\Delta u+u=f\) in \(\Omega\), \(u=g\) on \(\Gamma_ 1\), \(u_ n=b\) on \(\Gamma_ 2\), where \(\Omega\subset {\mathfrak R}^ 2\) is a polygonal domain \((\partial \Omega =\Gamma_ 1\cup \Gamma_ 2)\). Error estimates are also proved when the solution has singularities in the corners of the domain and in the case when g is not in the subspace of finite elements. It is mentioned that all conclusions are valid also for the elasticity problem. A numerical example (plain strain elasticity problem in an L-shaped domain) illustrates the applicability of the developed asymptotic theory.

Reviewer: J.Weisel

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05	Finite element methods applied to problems in solid mechanics
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B05	Classical linear elasticity
65N15	Error bounds for boundary value problems involving PDEs

Keywords:

h-p version of the finite element method; quasiuniform meshes; corner singularity; Error estimates; numerical example; plain strain elasticity problem; L-shaped domain; asymptotic

References:

[1]	I. BABUSKA and A. K. AZIZ, Survey Lectures on the Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), 3-359, Academic Press, New York, 1972. Zbl0268.65052 MR421106 · Zbl 0268.65052
[2]	I. BABUSKA, M. R. DORR, Error estimates for the combined h and p versions of finite element method. Numer. Math. 37 (1981), 252-277. Zbl0487.65058 MR623044 · Zbl 0487.65058
[3]	I. BABUSKA, W. GUI, B. GUO, B. A. SZABO, Theory and performance of the h-p version of the finite element method. To appear. Zbl0427.73067 · Zbl 0427.73067
[4]	I. BABUSKA, R. B. KELLOGG, J. PITKÄRANTA, Direct and inverse error estimates for finite element method. . SIAM J. Numer. Anal. 18 (1981), 515-545. Zbl0487.65059 · Zbl 0487.65059
[5]	I. BABUSKA, M. SURI, The optimal convergence rate of the p-version of the finite element method. Tech. Note BN-1045, Institute for Physical Science and Technology, University of Maryland, Oct. 1985. Zbl0637.65103 · Zbl 0637.65103
[6]	I. BABUSKA, B. A. SZABO and I. N. KATZ, The p-version of the finite element method. SIAM J. Numer. Anal. 18 (1981), 515-545. Zbl0487.65059 MR615529 · Zbl 0487.65059
[7]	I. BABUSKA and B. A. SZABO, On the rate of convergence of finite element method. Internat. J. Numer. Math. Engrg. 18 (1982), 323-341. Zbl0498.65050 MR648550 · Zbl 0498.65050
[8]	I. BERGH and J. LOFSTROM, Interpolation Spaces. Springer, Berlin, Heidelberg, New York, 1976. Zbl0344.46071 · Zbl 0344.46071
[9]	P. G. CIARLET, The Finite Element Method for Elliptic Problems. North-Holland, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[10]	M. R. DORR, The approximation theory for the p-version of the finite element method. SIAM J. Numer. Anal. 21 (1984), 1180-1207. Zbl0572.65074 MR765514 · Zbl 0572.65074
[11]	M. R. DORR, The Approximation of the Solutions of Elliptic Boundary-Value Problems via the p-Version of the Finite Element Method. SIAM J. Numer. Anal. 23 (1986), 58-77. Zbl0617.65109 MR821906 · Zbl 0617.65109
[12]	P. GRISVARD, Elliptic problems in nonsmooth domains. Pitman, Boston, 1985. Zbl0695.35060 MR775683 · Zbl 0695.35060
[13]	W. GUI and I. BABUSKA, The h, p and h-p versions of the finite element method for one dimensional problem : Part 1 : The error analysis of the p-version. Tech. Note BN-1036 ; Part 2 : The error analysis of the h and h-p versions. Tech. Note BN-1037 ; Part 3 : The adaptive h-p version, Tech. Note BN-1038, IPST, University of Maryland, College Park, 1985. To appear in Nume. Math.
[14]	B. GUO, I. BABUSKA, The h-p Version of the Finite Element Method. Part I : The basic approximation results. Part II : General results and applications. To appear in Comp. Mech. 1 (1986). Zbl0634.73059 MR1017747 · Zbl 0634.73059
[15]	G. H. HARDY, T. E. LITTLEWOOD, G. POLYA, Inequalities. Cambridge University Press, Cambridge, 1934. Zbl0010.10703 JFM60.0169.01 · Zbl 0010.10703
[16]	V. A. KONDRAT’EV, Boundary value problems for elliptic equations in domains with conic or angular points. Trans. Moscow Math. Soc. (1967), 227-313. Zbl0194.13405 MR226187 · Zbl 0194.13405
[17]	|17] J. L. LIONS, E. MAGENES, Non-homogeneous boundary value problems and applications-I. Springer-Verlag, Berlin, Heidelberg, New York, 1972. Zbl0223.35039 · Zbl 0223.35039
[18]	A. PINKUS, n-widths in Approximation Theory. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. Zbl0551.41001 MR774404 · Zbl 0551.41001
[19]	E. M. STEIN, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, N. J., 1970. Zbl0207.13501 MR290095 · Zbl 0207.13501
[20]	G. STRANG and G. J. FIX, An Analysis of the Finite Element Method. Prentice-Hall, Inglewood Cliffs, 1973. Zbl0356.65096 MR443377 · Zbl 0356.65096
[21]	B. A. SZABO, PROBE : Theoretical Manual. Noetic Technologies Corporation, St Louis, Missouri, 1985.
[22]	B. A. SZABO, Computation of Stress Field Parameters in Area of Steep stress gradients. Tech. Note WU/CCM-85/1, Center for Computational Mechanics, Washington University, 1985. Zbl0586.73170 · Zbl 0586.73170
[23]	B. A. SZABO, Mesh Design of the p-Version of the Finite Element Method. Lecture at Joint ASME/ASCE Mechanics Conference, Albuquerque, New Mexico, June 24-26, 1985. Report WV/CCM-85/2, Center for Computational Mechanics, Washington University, St Louis. Zbl0587.73106 · Zbl 0587.73106
[24]	B. A. SZABO, Implementation of a Finite Element Software System with h- and p-Extension Capabilities. Proc., 8th Invitational UFEM Symposium : Unification of Finite Element Software Systems. Ed. by H. Kardestuncer, The University of Connecticut, May 1985.