Convergence analysis of P1 finite element method for free boundary problems on non-overlapping subdomains

The paper applies a conforming P1 finite element method to general variational inequalities derived from free boundary problems. The domain of the free boundary problem can be properly split into two non-overlapping subdomains where the free boundary is located in only one subdomain. The variational inequality is reduced to a partial differential equation in the subdomain which does not contain the free boundary but still keeps its original form in the other subdomain. Therefore, the original variational inequality can be discretized separately with P1 finite element in different subdomains. A non-overlapping domain decomposition method is introduced to solve these two discretized sub-problems by P1 finite element method iteratively while a Robin type boundary condition is utilized for the data transfer on the common boundary. We show that the sequence of such finite element solutions converges to the discretized solution of the original problem. Application to a free seepage problem verifies the theory.