Mixed finite element approximation for non-divergence form elliptic equations with random input data

TL;DR

This paper develops a mixed finite element and collocation method for solving non-divergence form elliptic PDEs with random data, providing error analysis and numerical validation.

Contribution

It introduces a novel stochastic mixed formulation with mesh-dependent constraints and an uncoupled collocation approach for efficient computation.

Findings

Error bounds and convergence rates are established.

The method effectively handles randomness in coefficients and forcing.

Numerical results confirm theoretical predictions.

Abstract

We consider an elliptic partial differential equation in non-divergence form with a random diffusion matrix and random forcing term. To address this, we propose a mixed-type continuous finite element discretization in the physical domain, combined with a collocation discretization in the stochastic domain. For the mixed formulation, we first introduce a stochastic cost functional at the continuous level. This formulation is then enhanced to incorporate the vanishing tangential trace constraint directly into a mesh-dependent cost functional, rather than enforcing it in the solution's function space. In this context, we define a mesh-dependent norm and provide an error analysis based on this norm. We employ the collocation method by collocating the stochastic equation at the zeros of suitable tensor product orthogonal polynomials. This approach leads to a system of uncoupled deterministic problems, simplifying computation. Furthermore, we establish an a poriori error bound for the fully discrete approximation, detailing the convergence rates with respect to the discretization parameters. Finally, numerical results are presented to confirm and validate the theoretical findings.