Mesh modification for evolution equations

Finite element methods for which the underlying function spaces change with time are studied. The error estimates produced are all in norms that are very naturally associated with the problems. In some cases the Galerkin solution error can be seen to be quasi-optimal. K. Miller's moving finite element method is studied in one space dimension; convergence is proved for the case of smooth solutions of parabolic problems. Most, but not all, of the analysis is done on linear problems. Although second order parabolic equations are emphasized, there is also some work on first order hyperbolic and Sobolev equations.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  B. Wendroff,et al.  Generalized finite-difference schemes , 1969 .

[3]  J. Douglas,et al.  Galerkin Methods for Parabolic Equations , 1970 .

[4]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  T. Dupont.  Galerkin Methods for First Order Hyperbolics: An Example , 1973 .

[6]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[7]  P Jamet,et al.  A second order finite element method for the one‐dimensional Stefan problem , 1974 .

[8]  P Jamet,et al.  Numerical solution of the eulerian equations of compressible flow by a finite element method which follows the free boundary and the interfaces , 1975 .

[9]  Richard E. Ewing,et al.  Numerical Solution of Sobolev Partial Differential Equations , 1975 .

[10]  Pierre Jamet,et al.  Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain , 1977 .

[11]  P Jamet,et al.  Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements , 1977 .

[12]  Pierre Jamet.  Stability and Convergence of a Generalized Crank-Nicolson Scheme on a Variable Mesh for the Heat Equation , 1980 .

[13]  B. Finlayson,et al.  Oscillation limits for weighted residual methods applied to convective diffusion equations , 1980 .

[14]  V. Thomée,et al.  Maximum norm stability and error estimates in parabolic finite element equations , 1980 .

[15]  William G. Gray,et al.  Finite element simulation of flow in deforming regions , 1980 .

[16]  Bruce A. Finlayson,et al.  Solution of the transport equations using a moving coordinate system , 1980 .

[17]  Keith Miller,et al.  The moving finite element method: Applications to general partial differential equations with multiple large gradients☆ , 1981 .

[18]  Keith Miller,et al.  Moving Finite Elements. I , 1981 .

[19]  A Conservative Finite Element Method for One-Dimensional Stefan problems with Appearing and Disappearing Phases , 1981 .