Optimal error estimates of mixed FEMs for second order hyperbolic integro-differential equations with minimal smoothness on initial data

In this article, mixed finite element methods are discussed for a class of hyperbolic integro-differential equations (HIDEs). Based on a modification of the nonstandard energy formulation of Baker, both semidiscrete and completely discrete implicit schemes for an extended mixed method are analyzed and optimal L ∞ ( L 2 ) -error estimates are derived under minimal smoothness assumptions on the initial data. Further, quasi-optimal estimates are shown to hold in L ∞ ( L ∞ ) -norm. Finally, the analysis is extended to the standard mixed method for HIDEs and optimal error estimates in L ∞ ( L 2 ) -norm are derived again under minimal smoothness on initial data.

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

[2]  Yanping Lin,et al.  L 1 -ERROR ESTIMATES AND SUPERCONVERGENCE IN MAXIMUM NORM OF MIXED FINITE ELEMENT METHODS FOR NONFICKIAN FLOWS IN POROUS MEDIA , 2005 .

[3]  A. K. Pani,et al.  The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations , 1998 .

[4]  V. Thomée,et al.  Error estimates for some mixed finite element methods for parabolic type problems , 1981 .

[5]  W. H. Kirk.  And and Or , 1921 .

[6]  T. Dupont,et al.  A Priori Estimates for Mixed Finite Element Approximations of Second Order Hyperbolic Equations with , 1996 .

[7]  Richard E. Ewing,et al.  Sharp L2-Error Estimates and Superconvergence of Mixed Finite Element Methods for Non-Fickian Flows in Porous Media , 2002, SIAM J. Numer. Anal..

[8]  T. Dupont.  $L^2 $-Estimates for Galerkin Methods for Second Order Hyperbolic Equations , 1973 .

[9]  M. Wheeler,et al.  Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences , 1997 .

[10]  G. A. Baker.  Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations , 1976 .

[11]  Richard E. Ewing,et al.  Mixed Finite Element Approximations of Parabolic Integro-Differential Equations with Nonsmooth Initial Data , 2009, SIAM J. Numer. Anal..

[12]  Mary F. Wheeler,et al.  A Priori Error Estimates for Mixed Finite Element Approximations of the Acoustic Wave Equation , 2002, SIAM J. Numer. Anal..

[13]  Eric T. Chung,et al.  Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions , 2009, SIAM J. Numer. Anal..

[14]  Patrick Joly,et al.  An Analysis of New Mixed Finite Elements for the Approximation of Wave Propagation Problems , 2000, SIAM J. Numer. Anal..

[15]  R. K. Sinha.  Finite element approximations with quadrature for second‐order hyperbolic equations , 2002 .

[16]  Galerkin methods and L2-error estimates for hyperbolic integro-differential equations , 1989 .

[17]  Zhangxin Chen,et al.  Expanded mixed finite element methods for linear second-order elliptic problems, I , 1998 .

[18]  Sangita Yadav,et al.  Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data , 2013, J. Sci. Comput..

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

[20]  Michael Renardy,et al.  Mathematical problems in viscoelasticity , 1987 .

[21]  Vidar Thomée,et al.  Numerical methods for hyperbolic and parabolic integro-differential equations , 1992 .

[22]  T. Geveci.  On the application of mixed finite element methods to the wave equations , 1988 .

[23]  J. Rauch.  ON CONVERGENCE OF THE FINITE ELEMENT METHOD FOR THE WAVE EQUATION , 1985 .

[24]  Carol S. Woodward,et al.  Analysis of Expanded Mixed Finite Element Methods for a Nonlinear Parabolic Equation Modeling Flow into Variably Saturated Porous Media , 2000, SIAM J. Numer. Anal..