# Numerical Solutions of quasilinear-parabolic problems based on a space-time finite element scheme

## Dr. Ioannis Toulopoulos

**March 23, 2021, 2:30 p.m. ZOOM**

The description of real-life phenomena very often leads to second order parabolic problems of the form $u_t -\mathrm{div}\mathbf{A}(\nabla_x {u}) = f$, where $u_t:=\partial_t u$, $\nabla_x u$ is the spatial gradient of $u$ and $\mathbf{A}(\nabla_x {u}) = (\varepsilon +|\nabla_x u|)^{p-2}\nabla_x u$, with the parameters $\varepsilon \gt 0$ and $1 \lt p$. In general they can be seen as variations of the parabolic $p$-Laplacian problem where $\varepsilon=0$.

In this talk a continuous space-time finite element method will be analysed for approximating this class of quasilinear parabolic problems. The general ideas of the linear problems are used. We will introduce a space-time variational formulation where streamline upwind terms are further added for stabilizing the discretization in time direction. Then we will discuss some key points of the error analysis. In the last slides, a series of numerical tests will be presented that verify numerically the theoretical converge estimates. Emphasis will be given on the asymptotic convergence of the error parts which are related to the time discretization.

This work has been supported by the JKU-LIT project LIT-2017-4-SEE-004.