Even though a tremendous effort has been devoted to the area of transi
ent analysis, many of these models fall short of achieving the overall
desired objective due to oversights and rigid simplifying assumptions
. Some of those shortfalls are summarized as follows: (i) The grid use
d is too fine or too coarse and is usually fixed in size; (ii) No erro
r estimates are being used to trigger a grid size change; (iii) A fixe
d polynomial degree is being used to interpolate the approximate solut
ion of the differential equation at hand (finite element approach); (i
v) An approximate differential equation is used to model the physical
phenomena rather than the exact one (finite difference approach). In o
ur study, a self-adaptive grid refinement technique coupled with the p
-version of the finite element method is investigated. This type of ap
proach is called the hp-version of the finite element. In the modeling
process, the approximate solution to the exact differential equation
achieves convergence by applying two distinct solution enrichment stra
tegies. One strategy is to solve the problem using a very coarse grid
and to enrich the quality of the approximation by increasing the degre
e p of the interpolating polynomial shape functions for those elements
where it is needed. The other strategy involves local grid h refineme
nts for those elements where it is needed. Both strategies would inter
act synergistically to produce an optimal computational model that pro
duces an accurate simulation of the transient phenomena addressed with
minimal computational effort. The triggering for the increase or decr
ease in polynomial degree p and placement or removal of local mesh h r
efinements is based on the calculation of a reliable a posteriori erro
r estimate over each element at the end of each time-step. A case stud
y of particular interest is the simulation of wave propagation in a se
mi-infinite media.