I will present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities on an irregular domain. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L-infinty.

This lecture will describe the mathematical and statistical methods of estimating the errors in the optimal assessment of the past climate change, quantifying the uncertainties in the climate change detection, and analyzing the main uncertainty sources for climate predictions. Empirical orthogonal functions are extensively used to deal with spatial inhomogeneity. Temporal non-stationarity and model nonlinearity will be discussed. Detailed error analyses of the annual mean global and regional averages of the surface air temperature since 1861 will be presented.

This talk is dedicated to the studies of a three-dimensional, nonisothermal, anisotropic, two-phase transport model of proton exchange membrane fuel cell (PEMFC) and its efficient numerical method. Besides addressing the conservation equations of mass, momentum, species, charge and energy in view of the nonisothermality, anisotropy and multiphase flow in PEMFC model, from an efficient numerical method's point of view, we present some new formulations for species equations in the interests of the interactions among the species. In a framework of finite element-upwind finite volume method, some efficient numerical methods are designed and investigated in order to achieve fast and convergent numerical simulation for this PEMFC model.

Numerical implementation is done by using a novel automated finite element/finite volume program generator (FEPG). By virtue of a high-level algorithm description language (script), component programming and human intelligence technologies, FEPG can quickly generate finite element/finite volume source code for PEMFC simulation. Thus, one can focus on the efficient algorithm research without being distracted by the tedious computer programming on finite element/finite volume methods. The 3D numerical simulations demonstrate that a convergent and reasonable physical solution can be attained within 100 steps or so, comparing to the oscillating and nonconvergent nonlinear iterations conducted by the standard finite element/finite volume method. Numerical success demonstrates that FEPG is an efficient tool for both algorithm research and software development of a 3D multiphysics PEMFC model.