Hybrid-Trefftz Finite Elements for Non- homogeneous Parabolic Problems

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The solution of transient parabolic problems using a novel hybrid-Trefftz finite element is presented in this chapter. The governing equations are first discretised in time and reduced to a series of non-homogeneous elliptic problems in space variables only. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy the homogeneous form of the problem. Trefftz bases include regular functions of arbitrary orders and are defined independently for each finite element. 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. 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.