Mixed finite element approximation for non-divergence form elliptic equations with random input data
Amireh Mousavi
TL;DR
This paper develops a numerical framework for elliptic PDEs in non-divergence form with random input data by marrying a mixed finite element discretization in physical space with a stochastic collocation approach in parameter space. A mesh-dependent cost functional enables a stable, coercive discretization without enforcing tangential-trace constraints in the solution space, while collocation at tensor-product polynomial zeros yields decoupled deterministic systems for efficient computation. The authors establish an a priori error bound that decomposes spatial and stochastic contributions and validate the theory with numerical experiments showing exponential convergence in the stochastic dimension and favorable spatial rates. The methodology is particularly suited to problems with a low-dimensional random input and can handle Cordes-condition–driven well-posedness for non-divergence form operators.
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.