Discontinuous Galerkin method on reference domain

A reference domain is chosen to formulate numerical model using the discontinuous Galerkin with finite difference (DGFD) method. The differential problem, which is defined for the real domain, is transformed in a weak form to the reference domain. The shape of the real domain results from a considered problem which can be complex. On the other hand, a reference domain can be chosen to be, for example, cube or square, which is convenient for meshing and calculations. Transformation from the reference domain into the real one has to be defined. In this paper, the algorithm for such a transformation is proposed, which is based on second-order differential equations. The paper presents a series of benchmark examples that show both the correctness and flexibility of the proposed algorithms. In the majority of the examples, the reference domain is square when the real domains are, for example, quarter of annulus, circle or full annulus.