Hybrid-Trefftz finite elements for non-homogeneous parabolic problems using a novel dual reciprocity variant

A new hybrid-Trefftz finite element for the solution of transient, non-homogeneous parabolic problems is formulated. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases are regular, and defined independently for each finite element, using arbitrary orders. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same, regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. To enhance the reproducibility of this work, the formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community through the FreeHyTE web page.