Structure and numerical solution of a thermal problem with internal Dirichlet conditions

Inverse problems in geophysics seeking to understand the current state of planet Earth use data from multiple observables and involve a variety of physical principles. One of the key fields to determine is the temperature, affecting almost all the other physical quantities involved (e.g. densities, viscosities, wave propagation velocities, among others). The Lithosphere-Asthenosphere Boundary (LAB) is a boundary layer that affects most of the processes and properties in the Earth structure. Determining its location is one of the goals of inversions. It is usual within numerical studies to define the LAB as an isotherm. The need of determining the thermal field in accordance with a given LAB location leads to a mathematical problem with imposed interior Dirichlet conditions. In particular, the isotherm defining the LAB has to be located in the position tested by the inverse solver. Several approaches are sucessfully used to solve this kind of problem, but usually they lack of a sound physical model at least in some parts of the domain. Here we analyze the mathematical structure of a thermal problem with known interior conditions and then propose several numerical procedures to solve it. The proposed methods are tailored to the geophysical case and are based on the certainty of the different boundary conditions that are imposed in the model. Moreover, because this thermal solver is expected to be used many times within an inverse scheme, we want the numerical mesh to be fixed. The LAB, therefore, will not fit the mesh.