On high-order conservative finite element methods

Abstract

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...

On the Application of Stable Generalized Finite Element Method for Quasilinear Elliptic Two-Point BVP

“…16. This finding agrees with prior studies in [1]. As stated in this reference, the optimal convergence rate in L 2 norm can be recovered by adding the Lagrange multiplier as a corrector to the approximate solution.…”

“…An observation through numerical experiments reveals that using this technique, the optimal convergence property in H 1 semi-norm is preserved. An optimal convergence in L 2 -norm can be recovered by using the Lagrange multiplier values as a corrector (see [1]).…”

“…To get a way around this obstacle, we adopt a Lagrange multiplier technique introduced in [1,28]. The main idea lies on a recognition that typical linear variational formulation problem posed on a Banach space is equivalent to a minimization of a certain energy functional over that same space.…”

“…Another technique is called enriched Galerkin (EG) that is done by enriching the approximation space of the CG method with elementwise constant functions [33]. Yet another interesting approach was proposed in [1,28] that is conducted by constructing the approximate solution that combines the continuous Galerkin formulation and concurrently satisfies the local conservation restrictions. Procedures of this type utilize a Lagrange multiplier technique, where the approximation is viewed as a minimization of the energy functional over the finite element space under the constraint of algebraic representation of the local conservation property.…”

“…An error analysis in L 2 and H 1 spaces is carried out that confirms the optimality of SGFEM. Next, we employ the Lagrange multiplier technique as described in [1] to construct the SGFEM solution that satisfies the local conservation property. This is then validated by a numerical example showing that the optimal convergence behavior of the SGFEM is still maintained and at the same time the local conservation property is satisfied.…”