A family of $C^1$ finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems

— Families of C piecewise polynomial spaces of degree r ^ 3 on triangles and quadrilatéral in two dimensions are constructed, and approximation properties of the families are studied. Examples of the use of the families in Galerkin methodsfor 2nd and 4th order elliptic boundary value problems on arbitrarily shaped domains are given. The approximation properties on the boundary are such that the rate of convergence of the Galerkin methods is the optimal rate determined by the degree r of the piecewise polynomial space. Résumé. — En dimension deux, on construit des familles d'espaces de classe C .formés de polynômes de degré r ^ 3 par morceaux, sur des triangles et des quadrilatères, et on étudie les propriétés d'approximation de ces familles. On en donne des exemples d'application à des méthodes de Galerkin pour les problèmes aux limites elliptiques du 2 et du 4 ordre posés sur des domaines déforme arbitraire. Les propriétés d'approximation de la frontière sont telles que le taux de convergence des méthodes de Galerkin est le taux optimal, déterminé par le degré r de V'espace des polynômes par morceaux.

[1]  T. Dupont,et al.  Polynomial approximation of functions in Sobolev spaces , 1980 .

[2]  J. Blair.  Higher order approximations to the boundary conditions for the finite element method , 1976 .

[3]  P. Percell.  On Cubic and Quartic Clough–Tocher Finite Elements , 1976 .

[4]  L. R. Scott,et al.  A nodal basis for ¹ piecewise polynomials of degree ≥5 , 1975 .

[5]  R. Scott.  Interpolated Boundary Conditions in the Finite Element Method , 1975 .

[6]  Vidar Thomée,et al.  On Galerkin Methods in Semilinear Parabolic Problems , 1975 .

[7]  Jim Douglas,et al.  Collocation Methods for Parabolic Equations in a Single Space Variable , 1974 .

[8]  I. N. Katz,et al.  ADVANCED DESIGN TECHNOLOGY FOR RAIL TRANSPORTATION VEHICLES , 1974 .

[9]  A. H. Schatz,et al.  An observation concerning Ritz-Galerkin methods with indefinite bilinear forms , 1974 .

[10]  G. Strang.  Piecewise polynomials and the finite element method , 1973 .

[11]  J. Bramble,et al.  Rayleigh‐Ritz‐Galerkin methods for dirichlet's problem using subspaces without boundary conditions , 1970 .

[12]  J. Bramble,et al.  Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation , 1970 .

[13]  Gerd Grubb,et al.  PROBLÉMES AUX LIMITES NON HOMOGÉNES ET APPLICATIONS , 1969 .

[14]  E. H. Mansfield.  The Bending and Stretching of Plates , 1963 .

[15]  Jim DouglasJr.,et al.  H1-galerkin methods for a nonlinear dirichlet problem , 1977 .

[16]  Alberto Peano,et al.  Hierarchies of conforming finite elements for plane elasticity and plate bending , 1976 .

[17]  P. Grisvard,et al.  BEHAVIOR OF THE SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM IN A POLYGONAL OR POLYHEDRAL DOMAIN , 1976 .

[18]  P. G. Ciarlet,et al.  Numerical analysis of the finite element method , 1976 .

[19]  Ridgway Scott.  C1 Continuity via Constraints for 4th Order Problems , 1974 .

[20]  M. F. Wheeler,et al.  H1-Galerkin Methods for the Laplace and Heat Equations , 1974 .

[21]  P. G. Ciarlet,et al.  Sur l'élément de Clough et Tocher , 1974 .

[22]  J. F. Ciavaldini,et al.  Sur l'élément de Fraeijs de Veubeke et Sander , 1974 .

[23]  P. G. Ciarlet,et al.  General lagrange and hermite interpolation in Rn with applications to finite element methods , 1972 .

[24]  J. Nitsche,et al.  ON DIRICHLET PROBLEMS USING SUBSPACES WITH NEARLY ZERO BOUNDARY CONDITIONS , 1972 .

[25]  James H. Bramble,et al.  Least Squares Methods for 2mth Order Elliptic Boundary-Value Problems , 1971 .

[26]  James H. Bramble,et al.  Least squares methods for 2th order elliptic boundary-value problems , 1971 .