AN A PRIORI ERROR ANALYSIS OF A STRAIN GRADIENT MODEL USING <i>C</i><sup>0</sup> INTERIOR PENALTY METHODS

Abstract

In this work we study, from the numerical point of view, a strain gradient model. It can be written as a linear fourth-order in space and second-order in time partial differential equation which leads to a parabolic variational equation in terms of the velocity field. Then, a fully discrete approximation is provided by using the implicit Euler scheme to discretize the time derivatives and the so-called $ C^0 $ interior penalty method for the spatial approximation. A priori error estimates are obtained, and from them it follows the convergence of the approximations (under suitable regularity conditions). Finally, some two-dimensional numerical simulations are shown to demonstrate the numerical behaviour.