A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred. the phase in consideration (water, oil or gas); ( see e.g., [4,3,18,17,1,2]). The objective of find an approximation for 4 p satisfying the above equation and without loss of generality we assume Dirichlet boundary conditions. In general, 5 the forcing term q is due to gravity, sources or sinks. The mobility phase in consideration Λ(x) = K(x)k r (S (x))/µ, 6 where K(x) is absolute (intrinsic) permeability, k r is the relative phase permeability and µ the phase viscosity of the 7 fluid.Here Ω is a convex polygonal and two-dimesional domain with boundary ∂Ω. 8 Efficiently and accurately solving the equations like (1) governing fluid flow in oil reservoirs as well as in ground-9 water modeling and simulation of flow linked to advective/convective transport phenomena (e.g., [21,5]) is very 10 challenging because of the complex porous media environment and the intricate properties of fluid phases. A key 11 ingredient on the transport phenomena in porous media and related real-life applications is precisely the well-known 12 Darcy law, in which linked to equations in (1), is a fundamental PDE with a wide spectrum of relevance, of fundamen-13 tal applied mathematics [10,22], fundamental of modeling fluid flow flow through porous media [21,5] as well as 14 of a benchmark prototype model for proof-of-concept, efficient implementation and rigorous analysis for the design 15 and development of new finite element approaches, as the one discussed here, but also for other novel procedures, for 16 1 / Computers & Mathematics with Applications 00 (2016) 1-19 2 instance MsFEM [20], virtual finite elements [6], classical mixed finite elements [8]. Indeed, Discontinuous Galerkin 17 (DG) formulations have become an increasingly popular way to discretize the Darcy flow equations, either in the the 18 mixed finite element DG [11] or in the stabilized mixed DG [25] framework, just no name a few of the relevance 19 of model problem (1) from different perspectives. The field of fluid flow simulation in petroleum reservoirs [21] as 20well as the groundwater modeling and simulation of flow [5] linked to several transport phenomena have seen signif-21 icant advances in the last few decades (see, e.g., [28,16,14,13,12, 27]) due to novel discretizations associated do 22 Darcy problem (1), along with the challenges in modeling: flow and transport. We emphasize the challenges in the 23 construction of new methodologies into a reservoir simulation should have into account the following issues: 24• local mass...

In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be extended, without great difficulty, to more sophisticated GMsFEMs. For concreteness, the obtained error estimates generalize and simplify the convergence analysis of [J. Comput. Phys. 230 (2011), 937-955]. The GMsFEM method construct basis functions that are obtained by multiplication of (approximation of) local eigenvectors by partition of unity functions. Only important eigenvectors are used in the construction. The error estimates are general and are written in terms of the eigenvalues of the eigenvectors not used in the construction. The error analysis involve local and global norms that measure the decay of the expansion of the solution in terms of local eigenvectors. Numerical experiments are carried out to verify the feasibility of the approach with respect to the convergence and stability properties of the analysis in view of the good scientific computing practice.

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We study an efficient strategy based on finite elements to value spread options on commodities whose underlying assets follow a dynamic described by a certain class of two-dimensional Levy models by solving their associated partial integro-differential equation (PIDE). To this end we consider a Galerkin approximation in space along with an implicit θ-scheme for time evolution. Diffusion and drift in the associated operator are discretized using an exact Gaussian quadrature, while the integral part corresponding to jumps is approximated using the symbol method introduced in [1]. A system with blocked Toeplitz with Toeplitz blocks (BTTB) matrix is efficiently solved via biconjugate stabilized gradient method (BICSTAB) with a circulant pre-conditioner at each time step. The technique is applied to the pricing of crack spread options between the prices of futures RBOB gasoline (reformulated blendstock for oxygenate blending) and West Texas Intermediate(WTI) oil in NYMEX.

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