Coons-patch macroelements in two-dimensional parabolic problems

Having recently obtained encouraging results in elliptic and hyperbolic problems, this paper summarizes previous work and further investigates the performance of large isoparametric finite elements based on the Coons–Gordon interpolation formula in the analysis of two-dimensional parabolic potential problems. The latter formula allows the global interpolation of the potential within the whole problem domain and leads to the so-called Coons-patch-macroelements (CPM), where the degrees of freedom appear primarily at the element boundaries but in the general case it is also possible to use any desirable number of internal nodes. Mathematical and numerical aspects such as the relationship between boundary-only Coons-patch macroelements and Serendipity type elements, the systematic and straightforward way of adding internal nodes, the procedure of merging dissimilar domains and, finally, efficient numerical integration schemes are discussed. Numerical results on typical static (Laplace) and time-dependent thermal problems sustain the proposed method, which is successfully compared with conventional bilinear finite elements and exact analytical solutions.