Inversion of geophysical potential field data using the finite element method

The inversion of geophysical potential field data can be formulated as an optimization problem with a constraint in the form of a partial differential equation (PDE). It is common practice, if possible, to provide an analytical solution for the forward problem and to reduce the problem to a finite dimensional optimization problem. In an alternative approach the optimization is applied to the problem and the resulting continuous problem which is defined by a set of coupled PDEs is subsequently solved using a standard PDE discretization method, such as the finite element method (FEM). In this paper, we show that under very mild conditions on the data misfit functional and the forward problem in the three-dimensional space, the continuous optimization problem and its FEM discretization are well-posed including the existence and uniqueness of respective solutions. We provide error estimates for the FEM solution. A main result of the paper is that the FEM spaces used for the forward problem and the Lagrange multiplier need to be identical but can be chosen independently from the FEM space used to represent the unknown physical property. We will demonstrate the convergence of the solution approximations in a numerical example. The second numerical example which investigates the selection of FEM spaces, shows that from the perspective of computational efficiency one should use 2 to 4 times finer mesh for the forward problem in comparison to the mesh of the physical property.