FoCM 2014 conference

Workshop A5 - Multiresolution and Adaptivity in Numerical PDEs

December 11, 17:00 ~ 17:20 - Room B22

## Existence of $p$-Moments for the Weak Space-Time Heat Equation with Random Coefficients and Stability of its Petrov-Galerkin Discretization

### University of Cologne, Germany   -   cmollet@math.uni-koeln.de

We consider the heat equation in a weak space-time formulation with random right hand side and random spatial operator. The existence and uniqueness of a solution can be proven by the Banach-Necas-Babuska theorem. In this course we allow the spatial operator $A$ to have lower and upper bounds depending on a stochastic parameter $\omega$, i.e., we consider random variables $A_{\rm \min }(\omega)$ and $A_{\rm max}(\omega)$ as lower and upper bounds. The $L_p$-regularity of the solution and its connection to the random variables bounding $A$ can be proven. A similar approach is applied to a full space-time Petrov-Galerkin discretization. The stability of such an approach requires a lower bound for the discrete inf-sup condition independent of the grid spacing. Using similar ideas, we can prove stability when allowing a finer discretization for the test space than for the solution space.

Joint work with Stig Larsson (Chalmers University of Technology) and Matteo Molteni (Chalmers University of Technology).