Continuous Galerkin finite element methods for hyperbolic integro-differential equations

Abstract

Abstract. A hyperbolic integro-differential equation is considered, as a model problem, where the convolution kernel is assumed to be either smooth or no worse than weakly singular. Well-posedness of the problem is studied in the context of semigroup of linear operators, and regularity of any order is proved for smooth kernels. Energy method is used to prove optimal order a priori error estimates for the finite element spatial semidiscrete problem. A continuous space-time finite element method of order one is formulated for the problem. Stability of the discrete dual problem is proved, that is used to obtain optimal order a priori estimates via duality arguments. The theory is illustrated by an example.

Existence and convergence of Galerkin approximation for second order hyperbolic equations with memory term

“…Spatial finite element approximation of hyperbolic integro-differential equations similar to (1.1) has been studied in [10,12] and [13]. For some fully discrete methods see [3,7] and references therein.…”

“…For some fully discrete methods see [3,7] and references therein. In [10] and [12], for optimal order L ∞ (L 2 ) a priori error estimate for the solution u, they require two extra derivative of regularity of the solution.…”

“…In [10] and [12], for optimal order L ∞ (L 2 ) a priori error estimate for the solution u, they require two extra derivative of regularity of the solution. This was relaxed in [2] and [13] to one extra derivative, using the so called velocity-displacement form of the problem, that is, introducing the new variables u 1 = u and u 2 =u,…”

“…Here, we consider the spatial finite element semidiscretization of the problem as a system of second order ODEs, that can be combined with a temporal discretization method for such systems. Then using a different technique adapted from [15], for second order hyperbolic PDEs, we present an alternative proof to obtain L ∞ (L 2 ) optimal order a priori error estimate with one extra derivative regularity of the solution u, similar to [2] and [13].…”

Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

“…The convolution kernel is weakly singular and β ∈ L 1 (0, ∞) with ∞ 0 β(t) dt = γ. Well-posedness of the model problem (1.1) and more general form of such equations in fractional order viscoelasticity have been studied in [13], by means of Galerkin approximation methods. Continuous Galerkin methods of order one, both in time and space variables, have been applied to similar problems in [5], [11] and [12]. Discontinuous Galerkin and continuous Galerkin method, respectively, in time and space variables have been applied to a dynamic model problem in linear viscoelasticity (with exponential kernels) in [10].…”