Abstract
In this paper we investigatea priorierror estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.
Highlights
In this paper, we derive a priori error estimates for the space-time finite element discretization of a quasilinear gradient enhanced damage model
We investigate the finite element approximation of the quasilinear damage model
The model is motivated by a specific gradient enhanced damage model, first developed in [5, 6] and thoroughly analyzed for a mathematical point of view in [25, 26]
Summary
We derive a priori error estimates for the space-time finite element discretization of a quasilinear gradient enhanced damage model. Finite elements, quasilinear coupled PDE-ODE system, damage material model. There are contributions about the numerical analysis of parabolic equations, see for example [9,10,11,12,13] for uncontrolled equations or [24, 29] in the context of optimal control All of these works consider parabolic PDEs with a simpler structure than our damage model. The only contribution known to the author regarding numerical analysis of a damage model is the contribution [27], which considers a linearized, time-discrete phase-field model for crack propagation in the context of optimal control.
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