Abstract
This paper presents a robust and efficient Poisson solver that can produce accurate solutions and gradients (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic method for diffusion, where the Laplacian operator is discretized in the form of a hyperbolic system with solution gradients introduced as additional unknown variables. A practical formula for defining a reference length is proposed, which is needed to properly scale the relaxation length associated with the hyperbolic formulation for scale-invariant computations. The resulting system of residual equations is efficiently solved by a Jacobian-Free Newton-Krylov solver with an implicit defect-correction solver used as a variable-preconditioner. Robustness and superior gradient accuracy are demonstrated for linear and nonlinear Poisson equations through a series of numerical tests with unstructured tetrahedral grids.
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