The finite element method with nonuniform mesh sizes for unbounded domains
HTML articles powered by AMS MathViewer
- by C. I. Goldstein PDF
- Math. Comp. 36 (1981), 387-404 Request permission
Abstract:
The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem: \[ - \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty ,\] where ${\Omega ^C}$ is the complement in ${R^3}$ (three-dimensional Euclidean space) of a bounded set $\Omega$ with smooth boundary $\partial \Omega$, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary ${\Gamma _R}$ near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with ${\Omega ^C}$ is denoted by $\Omega _R^C$ and the given problem is replaced by \[ - \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}.\] This problem is then solved approximately by the finite element method, resulting in an approximate solution $u_R^h$ for each $h > 0$. In order to obtain a reasonably small error for $u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h)$, it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by $C{h^{ - 3}}$ with C independent of h and R.References
- Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
- Ivo Babuška, The finite element method for infinite domains. I, Math. Comp. 26 (1972), 1–11. MR 298969, DOI 10.1090/S0025-5718-1972-0298969-2
- J. Giroire and J.-C. Nédélec, Numerical solution of an exterior Neumann problem using a double layer potential, Math. Comp. 32 (1978), no. 144, 973–990. MR 495015, DOI 10.1090/S0025-5718-1978-0495015-8 S. Marin, A Finite Element Method for Problems Involving the Helmholtz Equation in Two Dimensional Exterior Regions, Thesis, Carnegie-Mellon University, Pittsburgh, Pa., 1978.
- A. H. Schatz and L. B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. I, Math. Comp. 32 (1978), no. 141, 73–109. MR 502065, DOI 10.1090/S0025-5718-1978-0502065-1
- Ivo Babuška, Finite element method for domains with corners, Computing (Arch. Elektron. Rechnen) 6 (1970), 264–273 (English, with German summary). MR 293858, DOI 10.1007/bf02238811
- S. C. Eisenstat and M. H. Schultz, Computational aspects of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 505–524. MR 0408269
- R. W. Thatcher, The use of infinite grid refinements at singularities in the solution of Laplace’s equation, Numer. Math. 25 (1975/76), no. 2, 163–178. MR 400748, DOI 10.1007/BF01462270 O. D. Kellogg, Foundations of Potential Theory, Ungar, New York, 1929.
- George Hsiao and R. C. MacCamy, Solution of boundary value problems by integral equations of the first kind, SIAM Rev. 15 (1973), 687–705. MR 324242, DOI 10.1137/1015093
- Martin Schechter, General boundary value problems for elliptic partial differential equations, Comm. Pure Appl. Math. 12 (1959), 457–486. MR 125323, DOI 10.1002/cpa.3160120305 A. Bayliss, M. Gunzberger & E. Turkel, Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions, ICASE Report 80-1, 1979.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 36 (1981), 387-404
- MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1981-0606503-5
- MathSciNet review: 606503