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We consider the numerical solution of elliptic problems in 3D with boundary nonlinearity, such as arising in stationary heat conduction models. We allow general non-orthotropic materials where the matrix of heat conductivities is a nondiagonal full matrix. The solution approach involves the finite element method (FEM) and Newton type iterations. We develop a quasi-Newton method for this problem, using spectral equivalence to approximate the derivatives. We derive the convergence of the method, and numerical experiments illustrate the robustness and the reduced computational cost.

Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity