Towards adaptive finite element schemes for partial differential Volterra equation solvers

We give a brief indication of how elliptic, parabolic and hyperbolic partial differential equations with memory arise when modelling viscoelastic materials. We then point out the urgent need for adaptive solvers for these problems and, employing the methodology of Eriksson, Johnson et al. (e.g., SIAM J. Numer. Anal. 28 (1991)), we given ana posteriori error estimate for a model two-point hereditary boundary value problem. The strengths and weaknesses of the analysis and estimate are discussed.

[1]  Jim Douglas,et al.  Numerical methods for integro-differential equations of parabolic and hyperbolic types , 1962 .

[2]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[3]  M. Huggins.  Viscoelastic Properties of Polymers. , 1961 .

[4]  G. Fairweather,et al.  Finite element methods for parabolic and hyperbolic partial integro-differential equations , 1988 .

[5]  Jace W. Nunziato,et al.  On heat conduction in materials with memory , 1971 .

[6]  Yanping Lin,et al.  A priori L 2 error estimates for finite-element methods for nonlinear diffusion equations with memory , 1990 .

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

[8]  M. K. Warby,et al.  Error Estimates with Sharp Constants for a Fading Memory Volterra Problem in Linear Solid Viscoelasticity , 1997 .

[9]  Vidar Thomée,et al.  Long-time numerical solution of a parabolic equation with memory , 1994 .

[10]  Simon Shaw,et al.  Discontinuous Galerkin method with a posteriori $L_p(0,t_i)$ error estimate for second-kind Volterra problems , 1996 .

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

[12]  J. Whiteman,et al.  Numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids , 1994 .

[13]  G. E. Troxell,et al.  Composition and properties of concrete , 1956 .

[14]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[15]  S. C. Hunter,et al.  Mechanics of Continuous Media , 1977 .

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

[17]  John M. Golden,et al.  Boundary Value Problems in Linear Viscoelasticity , 1988 .

[18]  Vidar Thomée,et al.  Time discretization of an integro-differential equation of parabolic type , 1986 .