A finite element method for elliptic problems with observational boundary data
For researchers solving PDEs with uncertain boundary data, this provides a theoretically grounded method with probabilistic error guarantees.
The paper proposes a finite element method for elliptic equations with noisy observational boundary data, proving convergence in expectation and exponential decay of error violation probability for sub-Gaussian noise.
In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show the convergence of the random finite element error in expectation and, when the noise is sub-Gaussian, in the Orlicz 2- norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.