Fardin Saedpanah

Stig Larsson, Milena Racheva, Fardin Saedpanah

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.

Fardin Saedpanah

A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions. Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity, under the appropriate assumptions on data.

Fardin Saedpanah

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.

Fardin Saedpanah

An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is presented such that it can be used for adaptive strategies based on dual weighted residual methods. A posteriori error estimates based on weighted global projections and local projections are also proved.