A low-dimensional approach for linear and nonlinear heat conduction in periodic domains

A low-order spectral method is used to solve steady-state linear and nonlin ear heat conduction problems with periodic boundary conditions and periodic geometry. The study consists of first mapping the complex geometry into a rectangular domain. The Galerkin projection method is applied to solve the mapped equations. It is found that a low number of modes usually are suffic ient to capture an accurate solution, Good agreement is obtained between th e low-order description and existing formulations. Both the finite element method (FEM and boundary element methods (BEM) are used for comparison.