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.
Keywords: Dual reciprocity method, Hybrid-Trefftz finite elemen, Particular
solution, Non-homogeneous parabolic problem.
