Quasi-optimality of Petrov-Galerkin discretizations of parabolic problems with random coefficients
Provides theoretical guarantees for numerical methods in stochastic parabolic PDEs, addressing a gap where random coefficients break standard assumptions.
The paper proves quasi-optimal error estimates for Petrov-Galerkin discretizations of parabolic problems with random coefficients, without assuming uniform bounds on coercivity and boundedness constants. It derives explicit inf-sup constants and establishes existence of solution moments.
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.