A compact multipoint flux approximation method with improved robustness

Multipoint flux approximation (MPFA) methods were introduced to solve control‐volume formulations on general grids. Although these methods are general in the sense that they may be applied to any grid, their convergence and monotonicity properties vary. We introduce a new MPFA method for quadrilateral grids termed the L‐method. This method seeks to minimize the number of entries in the flux stencils, while honoring uniform flow fields. The methodology is valid for general media. For homogeneous media and uniform grids in two dimensions, this method has four‐point flux stencils and seven‐point cell stencils, whereas the MPFA O‐methods have six‐point flux stencils and nine‐point cell stencils. The reduced stencil of the L‐method appears as a consequence of adapting the method to the closest neighboring cells, or equivalently, to the dominating principal direction of anisotropy. We have tested the convergence and monotonicity properties for this method and compared it with the O‐methods. For moderate grids, the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O‐methods is lost, while the L‐method converges with a reduced convergence rate. Also, the L‐method has a larger monotonicity range than the O‐methods. For homogeneous media and uniform parallelogram grids, the matrix of coefficients is an M‐matrix whenever the method is monotone. For strongly nonmonotone cases, the oscillations observed for the O‐methods are almost removed for the L‐method. Instead, extrema on no‐flow boundaries are observed. These undesired solutions, which only occur for parameters not common in applications, should be avoided by requiring that the previously derived monotonicity conditions are satisfied. For local grid refinements, test runs indicate that the L‐method yields almost optimal solutions, and that the solution is considerably better than the solutions obtained by the O‐methods. The efficiency of the linear solver is in many cases better for the L‐method than for the O‐methods. This is due to lower condition number and a reduced number of entries in the matrix of coefficients. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Convergence of the multipoint flux approximation L‐method for homogeneous media on uniform grids

In this article, we show the convergence of the multipoint flux approximation (MPFA) L‐method. To do so, we introduce a boundary modification of the original MPFA L‐method. On basis of this local modification, we obtain the equivalence between the MPFA L‐method and a conforming finite element approach with modified right hand side. The influence of the modified right hand side can be easily analyzed within the abstract framework of variational crimes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media

We propose a cell-centered symmetric scheme which combines the advantages of MPFA (multipoint flux approximation) schemes such as the L or the O scheme and of hybrid schemes: it may be used on general non-conforming meshes, it yields a 9-point stencil on two-dimensional quadrangular meshes, it takes into account the heterogeneous diffusion matrix, it is coercive and it can be shown to converge. The scheme relies on the use of special points, called harmonic averaging points, located at the interfaces of heterogeneity. To cite this article: L. Agelas et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A new smoothed aggregation multigrid method for anisotropic problems

A new prolongator is proposed for smoothed aggregation (SA) multigrid. The proposed prolongator addresses a limitation of standard SA when it is applied to anisotropic problems. For anisotropic problems, it is fairly standard to generate small aggregates (used to mimic semi‐coarsening) in order to coarsen only in directions of strong coupling. Although beneficial to convergence, this can lead to a prohibitively large number of non‐zeros in the standard SA prolongator and the corresponding coarse discretization operator. To avoid this, the new prolongator modifies the standard prolongator by shifting support (non‐zeros within a prolongator column) from one aggregate to another to satisfy a specified non‐zero pattern. This leads to a sparser operator that can be used effectively within a multigrid V‐cycle. The key to this algorithm is that it preserves certain null space interpolation properties that are central to SA for both scalar and systems of partial differential equations (PDEs). We present two‐dimensional and three‐dimensional numerical experiments to demonstrate that the new method is competitive with standard SA for scalar problems, and significantly better for problems arising from PDE systems. Copyright © 2008 John Wiley & Sons, Ltd.     

On the approximation method of solving integral equations in diffusion problems

In this paper we consider the one-dimensional problem of heat or mass transport in the system with moving ends. We show that without solving the heat transfer equation, the heat flux flowing out from the system can be found when temperature on the boundary of this system is known. We make use of the Banach contraction theorem for appropriate integral equations. Our method also enables us to find the distribution of temperature in the whole domain that forms the physical system.     

Analysis of a discontinuous galerkin method for heterogeneous diffusion problems with low‐regularity solutions

We study the convergence of the Symmetric Weighted Interior Penalty discontinuous Galerkin method for heterogeneous diffusion problems with low‐regularity solutions only belonging to W^{2, p} with p ∈ (1,2]. In 2d, we infer an optimal algebraic convergence rate. In any space dimension d, we achieve the same result for p > 2 d/( d + 2). We also prove convergence without algebraic rates for exact solutions only belonging to the energy space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

A posteriori error analysis of multipoint flux mixed finite element methods for interface problems

In this paper, the multipoint flux mixed finite element method is used to approximate the flux of two-dimensional elliptic interface problems. Within the class of modified quasi-monotonically distributed coefficients, we derive uniformly robust residual-type a posteriori error estimators for the flux error. Based on the residual-type estimator, we further develop robust implicit and explicit recovery-type estimators through gradient recovery in H(curl) conforming finite element spaces. Numerical experiments are presented to support the theoretical results.     

Convergence of the matrix transformation method for the finite difference approximation of fractional order diffusion problems

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R² and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments.     

On the homogenization of a diffusion–deformation problem in strongly heterogeneous media

We consider a porous fluid-saturated medium with periodic distribution of heterogeneities where the value of permeability decreases with the scale parameters. Homogenization of such double-porous material is performed using the method of periodic unfolding. The resulting homogenized macroscopic model is featured by the fading memory effect in the viscoelastic behaviour. This paper is based upon the work sponsored by the Ministry of Education of the Czech Republic under the research project MSM 49777513 03.     

On a method for construction of successive approximations for investigation of multipoint boundary-value problems

We suggest a new scheme of successive approximations. This scheme allows one to study the problem of existence and approximate construction of solutions of nonlinear ordinary differential equations with multipoint linear boundary conditions. This method enables one to study problems both with singular and nonsingular matrices in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1243–1253, September, 1995.     

在W12(R)空间中函数逼近的一种新方法

在再生核空间W12(R)中,利用再生核的性质实现了既不用计算导数也不需要计算积分,而只用函数值就可以将函数展开成级数的一种方法,并且这种级数的部分和{fn(x)}作为逼近f(x)的序列,它的误差rn(x)=f(x)-fn(x)在空间范数意义下单调下降.     

A plane wave least squares method for the Maxwell equations in anisotropic media

Numerical Algorithms - In this paper, we first consider the time-harmonic Maxwell equations with Dirichlet boundary conditions in three-dimensional anisotropic media, where the coefficients of the...     

A dual interpolation boundary face method with Hermite-type approximation for potential problems

This paper presents the dual interpolation boundary face method combined with a Hermite-type moving-least-squares approximation for solving complex two-dimensional potential problems. Compared to the standard algorithms, this combined method is better suited for structures with small feature sizes such as short edges and small chamfers. The interpolation functions, if constructed in cyclic coordinates, making it difficult to apply this new method to deal with complex structures with small feature sizes in which only one source point is assigned. The Hermite-type approximation formulated in Cartesian coordinates is able to completely overcome this obstacle by searching for source points on adjacent edges. Additionally, an improved and incomplete quadratic polynomial basis is presented to obtain an accurate algorithm for the Hermite-type approximation. We use several numerical examples to demonstrate the high accuracy and efficiency of the proposed method for solving various engineering structures with small feature sizes.     

A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations

Summary. The proposed method is based on an additive decomposition of the differential operator and the subsequent fitted discretization of the resulting components. For standard situations, the derived stability and error estimates in the energy norm qualitatively coincide with well-known estimates. In the case of small diffusion, a uniform error estimate with reduced order is obtained. Received August 7, 1997 / Revised version received July 15, 1998 / Published online December 6, 1999     

Averaging method in multipoint problems of the theory of nonlinear oscillations

By using the averaging method, we prove the solvability of multipoint problems for nonlinear oscillation systems and estimate the deviation of solutions of original and averaged problems.     

A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems

This paper proposes a new formulation of regularized meshless method (RMM), which differs from the traditional RMM in that the traditional formulation generates the diagonal elements of influence matrix via null-field integral equations, while our new one directly employs the boundary integral equations at the domain point to evaluate the diagonal elements. We test the present RMM formulation to two-dimensional anisotropic potential problems in finite and infinite domains in comparison with the traditional RMM. Numerical results show that the present RMM sharply outperforms the traditional RMM in the solution of interior problems, while the latter is clearly superior for exterior problems. A rigorous theoretical analysis of circular domain case also corroborates such numerical experiment observations and is provided in the appendix of this paper.     

Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media

We analyze family of solutions to multidimensional scalar conservation law, with flux depending on the time and space explicitly, regularized with vanishing diffusion and dispersion terms. Under a condition on the balance between diffusion and dispersion parameters, we prove that the family of solutions is precompact in L¹_loc{L^1_{\rm loc}}. Our proof is based on the methodology developed in Sazhenkov (Sibirsk Math Zh 47(2):431–454, 2006), which is in turn based on Panov’s extension (Panov and Yu in Mat Sb 185(2):87–106, 1994) of Tartar’s H-measures (Tartar in Proc R Soc Edinb Sect A 115(3–4):193–230, 1990), or Gerard’s micro-local defect measures (Gerard Commun Partial Differ Equ 16(11):1761–1794, 1991). This is new approach for the diffusion–dispersion limit problems. Previous results were restricted to scalar conservation laws with flux depending only on the state variable.