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A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

Let {Ah}h>0 be a family of elliptic finite element operators. Let I = [0, T] and consider the problem u′h(t) - Ahuh(t) = ƒh(t), t ∊ I, uh(0) = 0. In this paper, we show that for 1 < p < ∞ the solution of that problem satisfies the estimate ∥u′h∥Lp(I;Lp(Ω)) + ∥Ahuh∥Lp(I;Lp(Ω)) ≤ C∥ƒh∥Lp(I;Lp(Ω)), where C is independent of the parameter h and ƒh. In this case {Ah}h>0 is said to have discrete maximal Lp regularity. [ABSTRACT FROM AUTHOR]

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem. [ABSTRACT FROM AUTHOR]

We have recently developed a generic approach to solving retarded and neutral delay differential equations (DDEs). The approach is based on the use of an explicit continuous Runge­Kutta formula and employs defect control. The approach can be applied to problems with state-dependent delays and vanishing delays. In this paper we determine convergence properties for the numerical solution associated with methods that implement this approach. We first analyze such properties for retarded DDEs and then the analysis is extended to the case of neutral DDEs (NDEs). Such an extension is particularly important since NDEs have received little attention in the literature. The main result we establish is that the global error of the numerical solution can be efficiently and reliably controlled by directly monitoring the magnitude of the associated defect. Our analysis can also be applied directly to other numerical methods (which use defect error control) based on Runge­Kutta formulas for DDEs. [ABSTRACT FROM AUTHOR]