Uniqueness of finite element solutions to quasilinear PDE

Abstract: The finite element method is a popular and flexible framework for approximating solutions to partial differential equations. However, for nonlinear equations there remains much to be understood about properties of the discrete solutions, particularly on nonuniform meshes. Here we will investigate sufficient conditions for uniqueness of piecewise linear finite element solutions.

We will review some basic properties of the finite element discretization, then discuss a discrete comparison principle for a class of quasilinear PDE. As in the continuous setting, the comparison principle implies the uniqueness of the solution. We establish that without the presence of lower-order terms, it is sufficient for the meshsize to be locally controlled, based on the variance of the solution over each element, essentially requiring the mesh to be fine where the gradient is steep.