The present work studies the ®nite element approximation for the heat transfer process in an opaque three-dimensional plate with a temperature-dependent source dominating the conductive operator. The adopted mechanical model assumes the existence of a heat transfer from/to the plate following Newton's law of cooling. The numerical simulations performed have attested the instability of the classical Galerkin method when subjected to very high source-dominated regimen. Usual strategies in the Engineering practice of dealing with this shortcoming proved to be inef®cient. A Gradient-Galerkin/LeastSquares formulation was adopted in the numerical simulations as a remedy for the Galerkin's instability when subjected to those regimen. IntroductionIn most cases, ®nite element methods are based on Galerkin approximation, which, in the last decades, has been applied to approximate a large range of problems in Engineering (see, for instance, Bathe, 1982;Hughes, 1987; Pironneau, 1989 and references therein). Galerkin method was originally introduced for structural problems in which, assuming some restrictions usually present in engineering practical applications (Hughes and Franca, 1987;Franca et al., 1992), gives rise to symmetric elliptic operators and generates rather optimum convergence rates.However, when applied to¯uid problems, a slow development of the Galerkin approximation has been veri®ed. This fact may be explained by additional dif®culties that appear in applications such as incompressible¯ows, in which the pressure ®eld must be computed as a Lagrange multiplier of the velocity ®eld, thus generating a mixed problem which requires the compatibility of the ®nite element subspaces used to approximate the two above mentioned ®elds ± the so-called Babu ska-Brezzi condition (Babu ska, 1973; Brezzi, 1974). Besides, when the inertial term of the momentum equation is taken into account, the advective operator for advective-dominated ows is non-symmetric, leading to numerical pathologies in the Galerkin approximations of these problems such as the locking of the velocity ®eld and spurious oscillations in the pressure ®eld. Theses undesirable pathologies may be present even in such kinematical cases when a temperature-dependent heat source dominates the classical diffusive operator.The present work studies the ®nite element approximation of heat transfer process in an opaque three-dimensional plate (Saldanha da Gama, 1998) with a non uniform temperature-dependent source when this heat source dominates the conductive operator. The adopted mechanical model is obtained assuming the existence of a heat transfer from/to the plate following Newton's law of cooling. Besides, an integration of this model on plate thickness direction produces a two dimensional model in terms of a mean plate temperature. Numerical simulations of the above described problem have attested the instability inherent to Galerkin formulation in the presence of very high source-dominated regimen. Usual strategies in the Engineering practice to deal with th...

In this work, a model for the transport of a pollutant in the atmosphere is presented by considering mass and linear momentum conservation for the airpollutant mixture as well as the mass balance for the pollutant. The resulting mathematical description consists of a nonlinear system of hyperbolic equations that admits discontinuities in addition to smooth or classical solutions. The Riemann problem associated with a class of problems describing the transport of a pollutant in an ideal gas with constant temperature and discontinuous distribution for either mass density or velocity or pollutant concentration as initial condition is carefully discussed. Numerical approximations for this nonlinear system are obtained by employing Glimm's method and considered in some numerical simulations.

SUMMARYThis work studies an unsaturated ow of a Newtonian uid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian uid) and an inert gas (to account for the mixture compressibility). A set of two nonlinear partial di erential equations describes the problem, which is approximated by means of a Glimm's scheme, combined with an operator splitting technique.

No abstract

No abstract