# Conjugate Heat Transfer

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.