Numerical analysis of stochastic elliptic variational inequalities of second kind

Variational inequalities form an important family of nonlinear boundary value and initial-boundary value problems and they have been widely applied in science and engineering.

In the existing literature, variational inequalities are mostly studied in their deterministic form, which means that all the data are assumed to be known exactly. However, data in many real-world problems are only known approximately. As an example, when the data come from measurements, they are subject to measurement errors. Utilizing stochastic processes to model the data can be a useful approach to study such problems. Moreover, stochastic variational inequalities arise directly as the mathematical model in certain applications.

This thesis is primarily concerned with the numerical analysis of stochastic variational inequalities of the second kind with random coefficients or random noise. The finite element method is applied for the spatial discretization of the stochastic variational inequalities, and error estimates are derived for the numerical solution under certain solution regularity assumptions. The multi-level Monte Carlo approach is applied to estimate the expected value of the numerical solutions. Finally, numerical simulation results are reported on the performance of the numerical method. In particular, the numerical convergence order of the numerical solutions matches that of the theoretical prediction.