A Galerkin method for mixed parabolic–elliptic partial differential equations

In this article boundary value problems for partial differential equations of mixed elliptic–parabolic type are considered. To ensure that the considered problems possess a unique solution, the usual variational existence proof for parabolic problems is extended to the mixed situation. Further, the convergence of approximations computed by a time-space Galerkin method to the solution of the mixed problem is proven and error estimates are given.

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

[2]  Jinchao Xu,et al.  Regularity estimates for elliptic boundary value problems in Besov spaces , 2003, Math. Comput..

[3]  Endre Süli,et al.  hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. , 2002 .

[4]  G. Gustafson,et al.  Boundary Value Problems of Mathematical Physics , 1998 .

[5]  R. Rannacher.  Finite element solution of diffusion problems with irregular data , 1984 .

[6]  Heribert Blum,et al.  Algorithmic formulation and numerical implementation of coupled electromagnetic‐inelastic continuum models for electromagnetic metal forming , 2006 .

[7]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[8]  Paul Houston,et al.  Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems , 2001, SIAM J. Numer. Anal..

[9]  Dietrich Braess,et al.  Finite Elemente - Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie , 1992 .

[10]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[11]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

[12]  R. Arcangéli,et al.  Estimations des erreurs de meilleure approximation polynomiale et d'interpolation de Lagrange dans les espaces de Sobolev d'ordre non entier , 1984 .

[13]  Jacques-Louis Lions,et al.  Functional and variational methods , 1988 .

[14]  E. M. STEIN,et al.  THE CHARACTERIZATION OF FUNCTIONS ARISING AS POTENTIALS , 2007 .

[15]  R. Rannacher,et al.  On the Smoothing Property of the Galerkin Method for Parabolic Equations , 1982 .

[16]  J. Nédélec.  A new family of mixed finite elements in ℝ3 , 1986 .

[17]  E. Stein,et al.  The characterization of functions arising as potentials. II , 1961 .

[18]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems II: optimal error estimates in L ∞ L 2 and L ∞ L ∞ , 1995 .

[19]  Jacques-Louis Lions,et al.  Mathematical Analysis and Numerical Methods for Science and Technology: Volume 5 Evolution Problems I , 1992 .

[20]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems V: long-time integration , 1995 .

[21]  M. Fowler,et al.  Function Spaces , 2022 .