A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation

Numerical Methods Partial

2015

Self Cite

We study a second order hyperbolic initial-boundary value partial differential equation (PDE) with memory that results in an integro-differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example.

Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

Section: K(t − S)au(s) Ds = F (T) T ∈ (0 T ) With U(0)mentioning

confidence: 99%

See 1 more Smart Citation

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

Numerical Methods Partial

2015

Self Cite

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Larsson

Racheva

Computer Methods in Applied Mechanics and Engineering

2015

Self Cite

An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem. Stability estimates of the discrete problem are proved, that are used to prove optimal order a priori error estimates. The theory is illustrated by a numerical example.

“…From the extensive literature on theoritical and numerical analysis for partial differential equations with memory, we mention [13], [7], [2], [10], [14], and their references.…”

Section: Introductionmentioning

confidence: 99%

Continuous Galerkin finite element methods for hyperbolic integro-differential equations

IMA Journal of Numerical Analysis

2014

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.