Abstract
The conventional Galerkin finite element solution is mesh-dependent, and its discretization for Poisson's equation can not satisfy the conservation law over a nodal domain when unstructured linear meshes are used. This research tries to solve these problems by introducing a new concept of the virtual nodal domain(Vnd) for a linear quadrilateral element, and distributing the source term to a nodal algebraic equation in proportion to the area of the Vnd. The Vnd is evaluated using a second-order flux existing within a linear element. We proofed that the total Vnd of the four nodes equals to the area of the element, which guarantees that our scheme is also elementally conservative. Numerical simulation of heat conduction with both Dirichlet and Neumann boundary conditions shows that the accuracy has been improved obviously comparing with the conventional Galerkin FEM for unstructured quadrilateral meshes, especially for bad quality elements. Our scheme can be introduced into any commercial FEM code quite easily.
