Solving PDEs in non-rectangular 3D regions using a collocation finite element method

0965-9978/$ see front matter 2009 Elsevier Ltd. A doi:10.1016/j.advengsoft.2009.12.015 * Tel.: +1 915 747 6762. E-mail address: [email protected] The general-purpose partial differential equation (PDE) solver PDE2D uses a Galerkin finite element method, with standard triangular elements of up to fourth degree, to solve PDEs in general 2D regions. For 3D problems, a very different approach is used, which involves a collocation finite element method, with tricubic Hermite basis functions, and an automatic global coordinate transformation. If the user can define the 3D region by X 1⁄4 XðP1; P2; P3Þ; Y 1⁄4 YðP1; P2; P3Þ; Z 1⁄4 ZðP1; P2; P3Þ with constant limits on P1; P2; P3, then the PDEs and boundary conditions can be written in their usual Cartesian coordinate form and PDE2D will automatically convert the equations to the new coordinate system and solve the problem internally in this rectangle. The result is that for a wide range of simple 3D regions, once the global coordinate system is defined, the rest of the input is as simple as if the region were a rectangle. 2009 Elsevier Ltd. All rights reserved.