High resolution schemes for hyperbolic conservation laws

We develop a mixed finite element method for single phase flow in porous media that reduces to cell-centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients. Motivated by the multipoint flux approximation method where subedge fluxes are introduced, we consider the lowest order Brezzi-Douglas-Marini (BDM) mixed finite element method. A special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell-centered system for the pressures. Theoretical and numerical results indicate second-order convergence for pressures at the cell centers and first-order convergence for subedge fluxes. Second-order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular.

/pdf/a-multipoint-flux-mixed-finite-element-method-18glvw1075.pdf

A Multipoint Flux Mixed Finite Element Method

Superconvergence for the velocity along the Gauss lines is derived for the mixed finite element method for the second-order elliptic equation in rectangular domains. These estimates are based on the rectangular finite elements of Raviart–Thomas type.

Superconvergence of the velocity along the Gauss lines in mixed finite element methods

http://profdoc.um.ac.ir/articles/a/1070337.pdf

CFD analysis of natural gas emission from damaged pipelines: Correlation development for leakage estimation

This paper investigates different variants of the multipoint flux approximation (MPFA) O-method in 2D, which rely on a transformation to an orthogonal reference space. This approach yields a system of equations with a symmetric matrix of coefficients. Different methods appear, depending on where the transformed permeability is evaluated. Midpoint and corner-point evaluations are considered. Relations to mixed finite element (MFE) methods with different velocity finite element spaces are further discussed. Convergence of the MPFA methods is investigated numerically. For corner-point evaluation of the reference permeability, the same convergence behavior as the O-method in the physical space is achieved when the grids are refined uniformly or when grid perturbations of order h
 2 are allowed. For h
 2-perturbed grids, the convergence of the normal velocities is slower for the midpoint evaluation than for the corner-point evaluation. However, for rough grids, i.e., grids with perturbations of order h, contrary to the physical space method, convergence cannot be claimed for any of the investigated reference space methods. The relations to the MFE methods are used to explain the loss of convergence.

/pdf/convergence-of-a-symmetric-mpfa-method-on-quadrilateral-3e0bc4ppeb.pdf

Convergence of a symmetric MPFA method on quadrilateral grids

On the simulation of multicomponent gas flow in porous media

This paper establishes some superconvergence estimates for finite element solutions of second-order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution. The projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares. In particular, some local superconvergence estimates in the L2 and $L^\infty$ norms are derived for the local projections of the Galerkin finite element solution. The results have two prominent features. First, they are established for any quasi-uniform meshes, which are of practicalimportance in scientific computation. Second, they are derived on the basis of local properties of the domain and the solution for the second-order elliptic problem. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in practical computing.

An Interior Estimate of Superconvergence for Finite Element Solutions for Second-Order Elliptic Problems on Quasi-uniform Meshes by Local Projections

A new superconvergence result is established for numerical solutions of elliptic problems obtained from the mixed finite element method of Raviart--Thomas over rectangular partitions. The well-known optimal order error estimate in L2-norm for the flux approximation is of order ${\cal O}(h^{k+1})$, where $k\ge 0$ is the order of polynomials employed in the Raviart--Thomas element. The new superconvergence shows an improved accuracy of order ${\cal O}(h^{k+3})$ between the mixed finite element approximation and an appropriately defined local projection of the flux variable when k>0. A postprocessing technique using local projection methods is proposed and analyzed in order to provide a new approximate solution with the superconvergent order ${\cal O}(h^{k+3})$.

A New Superconvergence for Mixed Finite Element Approximations

A discretization scheme, which relates the mixed finite element method with cell-centered finite difference and finite volume element methods, is proposed for second-order elliptic equations on rectangular domains with locally refined composite grids. Optimal order error estimates and superconvergence results are established, both in L2 and pointwise along special Gauss-line loci. These error estimates hold for discontinuous piecewise constant conductivity under some special assumptions about solutions.

Point-distributed algorithms on locally refined grids for second order elliptic equations

A gradient recovery technique is proposed and analyzed for finite element solutions which provides new gradient approximations with high order of accuracy. The recovery technique is based on the method of least-squares surface fitting in a finite-dimensional space corresponding to a coarse mesh. It is proved that the recovered gradient has a high order of superconvergence for appropriately chosen surface fitting spaces. The recovery technique is robust, efficient, and applicable to a wide class of problems such as the Stokes and elasticity equations.

Superconvergence for the Gradient of Finite Element Approximations by L 2 Projections

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