Quasi-optimality of Petrov-Galerkin discretizations of parabolic problems with random coefficients

TL;DR

This paper analyzes the stability and error estimates of Petrov-Galerkin discretizations for parabolic problems with random coefficients, providing explicit formulas and moment bounds without uniform coercivity assumptions.

Contribution

It introduces a weak space-time formulation for parabolic problems with random operators and derives quasi-optimal error estimates and explicit inf-sup constants.

Findings

Existence of moments for the solution under random coefficients

Explicit formulas for inf-sup constants in the weak formulation

Quasi-optimal error bounds for Petrov-Galerkin discretizations

Abstract

We consider a linear parabolic problem with random elliptic operator in the usual Gelfand triple setting. We do not assume uniform bounds on the coercivity and boundedness constants, but allow them to be random variables. The parabolic problem is studied in a weak space-time formulation, where we can derive explicit formulas for the inf-sup constants. Under suitable assumptions we prove existence of moments of the solution. We also prove quasi-optimal error estimates for piecewise polynomial Petrov-Galerkin discretizations.