Computation with adaptive grid refinement has proved to be a useful and efficient tool in scientific computing over the last several decades. The key behind this technique is the design of a good a posterior error estimator that provides a guidance on how and where grids should be refined. In this paper, the authors propose and analyze a posteriori error estimator for a stabilized finite element method in computational fluid dynamics. The main contributions of the paper are: (1) an efficient a posteriori error estimator is designed and analyzed for a general stabilized finite element method, (2) a rigorous mathematical analysis is established for a theoretical justification of its efficiency and generality to other applications, and (3) some computational results with a comparison with other methods are presented for a computational justification of the proposed a posteriori error estimator.

A new multiphysics model for two-phase compositional flow is presented. It is designed to fit the level of model complexity to that of the flow and transport processes taking place in a given region of the domain. Thus, the model domain is divided into a subdomain which accounts for two-phase compositional processes and another in which single-phase transport is described. A coupling of the simple and complex equations gives rise to an efficient model. Special interest is placed in the discretization of the two-phase compositional model in a finite-volume context and an IMPES time scheme with decoupled pressure and transport equations. For optimal subdomain determination, an easy-to-handle, adaptive scheme is presented. The practical usability is demonstrated on a real live problem from carbon dioxide sequestration.

We develop and analyze an immersed Eulerian-Lagrangian localized adjoint method (ImELLAM) for transient advection-diffusion equations with interfaces. The derived method possesses the combined advantages of the immersed finite element method and the Eulerian-Lagrangian method.

We present a characteristic method for the solution of the transient advection diffusion equations in two space-dimensions. This method uses Wachspress-type rational basis functions over hexagonal grids within the framework of the Eulerian Lagrangian localized adjoint methods (ELLAM). It therefore maintains the advantages of previous ELLAM schemes and generates accurate numerical solutions even if large time steps are used in the simulation. Numerical experiments are presented to illustrate the performance of this method and to investigate its convergence numerically.

We prove an optimal-order error estimate in a degenerate-diffusion weighted energy norm for implicit Euler and Crank-Nicolson finite difference methods to two-dimensional time-dependent advection-diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not on the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal-order estimate of the finite difference methods, in which the generic constants depend only on the Sobolev norms of the initial and right-hand side data.

We prove an optimal-order error estimate in a weighted energy norm for bilinear Galerkin finite element method for two-dimensional time-dependent advection-diffusion equations by the means of integral identities or expansions, in the sense that the generic constants in the estimates depend only on certain Sobolev norms of the true solution but not on the scaling parameter $\varepsilon$. These estimates, combined with a priori stability estimates of the governing partial differential equations, yield an "$\varepsilon$-uniform estimate of the bilinear Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right data but not on the scaling parameter $\varepsilon$.

For the transient behavior of a semiconductor device, the modified method of characteristics finite element alternating-direction procedures with moving meshes are put forward. Some techniques, such as calculus of variations, operator-splitting, characteristic method, generalized $L^2$ projection, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in $L^2$ norm are derived for the error in the approximation solution. Thus the well-known theoretical problem has been thoroughly and completely solved.

In this article we construct an approximate similarity solution to a nonlinear diffusion equation in spherical coordinates. In hydrology this equation is known as the Boussinesq equation when written in planar or cylindrical coordinates. Recently Li et al. [8] obtained an approximate similarity solution to the Boussinesq equation in cylindrical coordinates. Here we consider the same problem in spherical coordinates with the prescribed power law point source boundary condition. The resulting scaling function has a power law singularity at the origin versus a logarithmic singularity in the cylindrical case.

In this paper, we present the analysis of recently introduced multiscale finite element methods that employ limited global information. In particular, these methods use single-phase flow information for the construction of more accurate solution for two-phase immiscible flow dynamics in heterogeneous porous media. We consider the analysis of Galerkin multiscale finite element method as well as mixed multiscale finite element method. Our analysis assumes that the fine-scale features of two-phase flow dynamics strongly depend on single-phase flow. Under this assumption, we present the analysis of multiscale finite element methods that use single-phase flow information. Numerical results are presented which demonstrate that MsFEM using limited global information is more accurate and converges as the coarse mesh size decreases.

We present an $H^1$-Galerkin mixed finite element method for the solution of a nonlinear parabolic pressure equation, which arises in the mathematical models for describing a compressible fluid flow process in subsurface porous media. The method possesses the advantages of mixed finite element methods while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. We conducted theoretical analysis to study the existence and uniqueness of the numerical solutions of the scheme and prove an optimal-order error estimate for the method. Numerical experiments are performed to justify the theoretical analysis.

A modification of the modified method of characteristics (MMOC) is introduced for solving the coupled system of partial differential equations governing miscible displacement in porous media . The pressure-velocity is approximated by a mixed finite element procedure using a Raviart-Thomas space of index $k$ over a uniform grid. The resulting Darcy velocity is post-processed by convolution with Bramble-Schatz kernel and this enhanced velocity is used in the evaluation of the coefficients in MMOC for the concentration equation. If the concentration space is of local degree $l$, then, the error in the concentration is $O(h^{l+1}_c+h^{2k+2}_p)$, which reflects the superconvergence of velocity approximation.