On the statement and numerical solution of the thermal problem within inversion methods for the study of lithospheric structure.

One of the goals of geophysicists is mapping and understanding the current structure of the Earth including its variations in composition, temperature and dynamical state. This structure is only accessible via indirect observations and, therefore, the mathematical problem to be solved is of an inverse kind. Within the inverse solver, many forward problems will be tested until finding a configuration compatible with the observations. This work deals with the problem statement and numerical solution of the forward thermal problem that arises from an inverse solver. In this case, we will use a simple parameterization of the Lithosphere-Asthenosphere Boundary (LAB), but the results are useful for other parametric description (e.g. one parameter per each cell). 
A simplified model is used to show the ill-posedness of the mathematical problem arising when the LAB --an isotherm whose location is determined by the input parameters-- is imposed within the domain, over-constraining the forward problem. This is well-known in the community and several authors have proposed different approaches to circumvent it. Nevertheless, the strategies used in practice usually involve some non-physical procedures such as transitional regions where two different temperature fields are made compatible by smearing out differences. Generally, the solution in these regions does not comply with the governing equation and exhibits a non-physical behaviour. 
In this work, we propose a specific problem statement for the temperature with interior essential conditions. The resulting problem is mathematically sound and results in a two-step numerical solver. This guarantees a self-consistent temperature field, in the sense that it respects the thermal governing equations everywhere. 
The numerical domain is divided into two subdomains (lithosphere and asthenosphere) that are solved separately in the same mesh, using an unfitted mesh methodology. First, the temperature of the lithosphere is computed using the essential condition on the LAB. Second, the temperature in the mantle is obtained by minimizing a residual that measures the compatibility between the two subdomains in terms of LAB temperatures and across-LAB fluxes. This is done by adjusting the proper fluxes at the bottom of the numerical domain. 
Several examples are presented showing that the obtained temperature fields are stable and oscillation-free. Moreover, the resulting fluxes at the bottom of the domain are reasonable and compatible with the expected values.