High-accuracy finite-element methods for positive symmetric systems

A nonstandard-type “least'squares” finite-element method is proposed for the solution of first-order positive symmetric systems. This method gives optimal accuracy in a norm similar to the H1 norm. When a regularity condition holds it is optimal in L2 as well. Otherwise, it gives errors suboptimal by only h12 (where h is the mesh diameter). Thus, it has greater accuracy than usual finite-element, finite-difference or least-squares methods for such problems. In addition, the spectral condition number of the associated linear system is only O(h−1) vs. O(h−2) for the usual least-squares methods. Thus, the method promises to be an efficient, high-accuracy method for hyperbolic systems such as Maxwell's equations. It is also equally promising for mixed-type equations that have a formulation as a positive symmetric system.

[1]  T. Katsanis.  Numerical Solution of Tricomi Equation Using Theory of Symmetric Positive Differential Equations , 1969 .

[2]  Max Gunzburger,et al.  On mixed finite element methods for first order elliptic systems , 1981 .

[3]  GALERKIN METHODS FOR TWO-POINT BOUNDARY VALUE PROBLEMS FOR FIRST ORDER SYSTEMS* , 1983 .

[4]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[5]  P. Lesaint.  Finite element methods for the transport equation , 1974 .

[6]  T. Hughes,et al.  Streamline upwind formulations for advection-diffusion, Navier-Stokes, and first-order hyperbolic equations. , 1982 .

[7]  G. D. Raithby,et al.  Skew upstream differencing schemes for problems involving fluid flow , 1976 .

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

[9]  Kurt Friedrichs,et al.  On Symmetrizable Differential Operators , 1986 .

[10]  T. Katsanis.  Numerical Solution of Symmetric Positive Differential Equations , 1968 .

[11]  J. E. Dendy.  Two Methods of Galerkin Type Achieving Optimum $L^2 $ Rates of Convergence for First Order Hyperbolics , 1974 .

[12]  Vassilios A. Dougalis,et al.  The Effect of Quadrature Errors on Finite Element Approximations for Second Order Hyperbolic Equations , 1976 .

[13]  L. Wahlbin,et al.  A Dissipative Galerkin Method for the Numerical Solution of First Order Hyperbolic Equations , 1974 .

[14]  Antony Jameson,et al.  NUMERICAL SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF MIXED TYPE , 1976 .

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

[16]  Stanley Osher,et al.  A Finite Element Method for a Boundary Value Problem of Mixed Type , 1979 .

[17]  Ernst P. Stephan,et al.  Remarks to Galerkin and least squares methods with finite elements for general elliptic problems , 1976 .

[18]  Max Gunzburger,et al.  On least squares approximations to indefinite problems of the mixed type , 1978 .

[19]  The effect of numerical integration in finite element approximations to degenerate evolution equations , 1984 .

[20]  A. K. Aziz,et al.  Finite Element Approximation for First Order Systems , 1978 .

[21]  Claes Johnson,et al.  Finite element methods for linear hyperbolic problems , 1984 .

[22]  P. Lesaint,et al.  Finite element methods for symmetric hyperbolic equations , 1973 .

[23]  Estimates away from a discontinuity for dissipative Galerkin methods for hyperbolic equations , 1981 .

[24]  Rudolf Mathon,et al.  Least-Squares Methods for Mixed-Type Equations , 1981 .

[25]  K. Friedrichs.  Symmetric hyperbolic linear differential equations , 1954 .

[26]  M. Gurtin,et al.  On patched variational principles with application to elliptic and mixed elliptic-hyperbolic problems , 1977 .