A least-squares approach based on a discrete minus one inner product for first order systems

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in H -1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.

[1]  L. R. Scott,et al.  Simultaneous approximation in scales of Banach spaces , 1978 .

[2]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[3]  J E Pasciak,et al.  First-order System Least- Squares for Second-order Partial Diierential Equations: Part Ii. Siam J , 1996 .

[4]  P. Vassilevski,et al.  Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates , 1996 .

[5]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: II. Beyond SUPG , 1986 .

[6]  G. F. Carey,et al.  Least-Squares Mixed Finite Element Methods for Non-Selfadjoint Elliptic Problems: II. Performance of Block-ILU Factorization Methods , 1995, SIAM J. Sci. Comput..

[7]  M. Dauge.  Elliptic boundary value problems on corner domains , 1988 .

[8]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .

[9]  Dennis C. Jespersen,et al.  A least squares decomposition method for solving elliptic equations , 1977 .

[10]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[11]  Pekka Neittaanmäki,et al.  On finite element approximation of the gradient for solution of Poisson equation , 1981 .

[12]  T. Manteuffel,et al.  FIRST-ORDER SYSTEM LEAST SQUARES FOR SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS : PART II , 1994 .

[13]  Randolph E. Bank,et al.  An optimal order process for solving finite element equations , 1981 .

[14]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[15]  Pavel B. Bochev,et al.  Accuracy of Least-Squares Methods For , 1993 .

[16]  Graham F. Carey,et al.  Convergence studies of least-squares finite elements for first-order systems , 1989 .

[17]  Tsu-Fen Chen,et al.  On least-squares approximations to compressible flow problems , 1986 .

[18]  Ching L. Chang,et al.  Finite element approximation for grad-div type systems in the plane , 1992 .

[19]  Louis A. Povinelli,et al.  Optimal least-squares finite element method for elliptic problems , 1993 .

[20]  G. Carey,et al.  Least-squares mixed finite elements for second-order elliptic problems , 1994 .

[21]  M. D. Gunzburger,et al.  Least Squares Approximations to Compressible Flow Problems , 1984 .

[22]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[23]  W. Wendland.  Elliptic systems in the plane , 1979 .

[24]  G. Fix,et al.  Least squares finite element simulation of transonic flows , 1986 .

[25]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[26]  R. Bank,et al.  5. Variational Multigrid Theory , 1987 .

[27]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[28]  R. B. Kellogg,et al.  Least Squares Methods for Elliptic Systems , 1985 .

[29]  M. Gunzburger,et al.  Accuracy of least-squares methods for the Navier-Stokes equations , 1993 .

[30]  B. Jiang,et al.  Least-square finite elements for Stokes problem , 1990 .

[31]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[32]  Thomas J. R. Hughes,et al.  A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems , 1986 .

[33]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

[34]  M. Gunzburger,et al.  Analysis of least squares finite element methods for the Stokes equations , 1994 .

[35]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .

[36]  J. Pasciak,et al.  New convergence estimates for multigrid algorithms , 1987 .

[37]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

[38]  M. Dauge.  Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions , 1988 .