Molnar and England (1990, https://doi.org/10.1029/JB095iB04p04833) introduced equations using a semianalytical approach that approximate the thermal structure of the forearc regions in subduction zones. A detailed new comparison with high‐resolution finite element models shows that the original equations provide robust predictions and can be improved by a few modifications that follow from the theoretical derivation. The updated approximate equations are shown to be quite accurate for a straight‐dipping slab that is warmed by heat flowing from its base and by shear heating at its top. The approximation of radiogenic heating in the crust of the overriding plate is less accurate but the overall effect of this heating mode is small. It is shown that the previous and updated approximate equations become increasingly inaccurate with decreasing thermal parameter and increasing variability of slab dip. It is also shown that the approximate equations cannot be extrapolated accurately past the brittle‐ductile transition. Conclusions in a recent paper (Kohn et al., 2018, https://doi.org/10.1073/pnas.1809962115) that modest amount of shear heating can explain the thermal conditions of past subduction from the exhumed metamorphic rock record are invalid due to a number of compounding errors in the application of the Molnar and England (1990, https://doi.org/10.1029/JB095iB04p04833) equations past the brittle‐ductile transition. The use of the improved approximate equations is highly recommended provided their limitations are taken into account. For subduction zones with variable dip and/or low thermal parameter finite element modeling is recommended.

The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Fréchet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems.

Nearly all chemical heterogeneity in the mantle is the result of geological processes that have modified its initial composition. The one exception is heterogeneity that exists because some portion of the mantle remains unmodified. This "primitive" material has been identified by measuring noble gas concentrations of modern mantle derived rock (

No abstract

No abstract