A new alternating‐direction finite element method for hyperbolic equation

A modified backward difference time discretization is presented for Galerkin approximations for nonlinear hyperbolic equation in two space variables. This procedure uses a local approximation of the coefficients based on patches of finite elements with these procedures, a multidimensional problem can be solved as a series of one‐dimensional problems. Optimal order H₀¹ and L² error estimates are derived. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A locally conservative Eulerian‐Lagrangian control‐volume method for transient advection‐diffusion equations

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection‐diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well‐regarded Eulerian‐Lagrangian methods, which were previously shown to be very competitive with many well‐perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

Alternating direction finite volume element methods for 2D parabolic partial differential equations

On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L² norm or H¹ semi‐norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A discontinuous finite volume element method for second‐order elliptic problems

In this article, we propose a new discontinuous finite volume element (DFVE) method for the second‐order elliptic problems. We treat the DFVE method as a perturbation of the interior penalty method and get a superapproximation estimate in a mesh dependent norm between the solution of the DFVE method and that of the interior penalty method. This reveals that the DFVE method is much closer to the interior penalty method than we have known. By using this superapproximation estimate, we can easily get the optimal order error estimates in the L² ‐norm and in the maximum norms of the DFVE method.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 425–440, 2012     

A new combined finite element‐upwind finite volume method for convection‐dominated diffusion problems

In this article, we develop a combined finite element‐weighted upwind finite volume method for convection‐dominated diffusion problems in two dimensions, which discretizes the diffusion term with the standard finite element scheme, and the convection and source terms with the weighted upwind finite volume scheme. The developed method leads to a totally new scheme for convection‐dominated problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high‐order accuracy. Stability analyses of the scheme are given for the problems with constant coefficients. Numerical experiments are presented to illustrate the stability and optimal convergence of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 799–818, 2016     

A uniformly optimal‐order estimate for finite volume method for advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for finite volume method for two‐dimensional time‐dependent advection–diffusion equations on a uniform space‐time partition of the domain. The generic constants in the estimates depend only on certain norms of the true solution but not on the scaling parameter. These estimates, combined with a priori stability estimates of the governing partial differential equations with full regularity, yield a uniform estimate of the finite volume method, in which the generic constants depend only on the Sobolev norms of the initial and right side data but not on the scaling parameter. We use the interpolation of spaces and stability estimates to derive a uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right‐hand side data. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 17‐43, 2014     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Analysis and efficient implementation of alternating direction implicit finite volume method for Riesz space‐fractional diffusion equations in two space dimensions

In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has memory requirement and computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations.     

A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

We present a parallel matrix‐free implicit finite volume scheme for the solution of unsteady three‐dimensional advection‐diffusion‐reaction equations with smooth and Dirac‐Delta source terms. The scheme is formally second order in space and a Newton–Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix‐vector product required is hardcoded without any approximations, obtaining a matrix‐free method that needs little storage and is well‐suited for parallel implementation. We describe the matrix‐free implementation of the method in detail and give numerical evidence of its second‐order convergence in the presence of smooth source terms. For nonsmooth source terms, the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long‐time simulation of calcium flow in heart cells and show its parallel scaling. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 143–167, 2015     

The discrete duality finite volume method for a class of quasi‐Newtonian Stokes flows

In this paper, we propose a discrete duality finite volume (DDFV) scheme for the incompressible quasi‐Newtonian Stokes equation. The DDFV method is based on the use of discrete differential operators which satisfy some duality properties analogous to their continuous counterparts in a discrete sense. The DDFV method has a great ability to handle general geometries and meshes. In addition, every component of the velocity gradient can be reconstructed directly, which makes it suitable to deal with the nonlinear terms in the quasi‐Newtonian Stokes equation. We prove that the proposed DDFV scheme is uniquely solvable and of first‐order convergence in the discrete L²‐norms for the velocity, the strain rate tensor, and the pressure, respectively. Ample numerical tests are provided to highlight the performance of the proposed DDFV scheme and to validate the theoretical error analysis, in particular on locally refined nonconforming and polygonal meshes.     

The characteristic finite element alternating direction method with moving meshes for nonlinear convection‐dominated diffusion problems

In modern numerical simulation of prospecting and exploiting oil‐gas resources and in environmental science, it is necessary to consider numerical method of nonlinear convection‐dominated diffusion problems. This thesis, starting from actual conditions such as the three‐dimensional characteristics of large‐scale science‐engineering computation, puts forward a kind of characteristic finite element alternating direction method with moving meshes. Some techniques, such as calculus of variations, operator‐splitting, generalized L² projection, energy method, negative norm estimate, the theory of prior estimates and techniques, are adopted. Optimal order estimates in L² norm are derived to determine the errors in the approximate solution. Thus the important theoretical problem has been solved. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A space‐time mixed‐hybrid finite element method for the damped wave equation

A space‐time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed‐hybrid finite element methods, and where an original conforming approximation of H(div;Ω) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Convergence of an implicit Voronoi finite volume method for reaction–diffusion problems

We investigate the convergence of an implicit Voronoi finite volume method for reaction–diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical activities of the involved species as primary variables. The numerical scheme works with boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh‐independent global upper, and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. To illustrate the preservation of qualitative properties by the numerical scheme, we present a long‐term simulation of the Michaelis–Menten–Henri system. Especially, we investigate the decay properties of the relative free energy over several magnitudes of time, and obtain experimental orders of convergence for this quantity. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 141–174, 2016     

Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Superconvergence and gradient recovery for a finite volume element method for solving convection‐diffusion equations

We study the superconvergence of the finite volume element (FVE) method for solving convection‐diffusion equations using bilinear trial functions. We first establish a superclose weak estimate for the bilinear form of FVE method. Based on this estimate, we obtain the H¹‐superconvergence result: . Then, we present a gradient recovery formula and prove that the recovery gradient possesses the ‐order superconvergence. Moreover, an asymptotically exact a posteriori error estimate is also given for the gradient error of FVE solution.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1152–1168, 2014     

Study of the mixed finite volume method for Stokes and Navier‐Stokes equations

We present finite volume schemes for Stokes and Navier‐Stokes equations. These schemes are based on the mixed finite volume introduced in (Droniou and Eymard, Numer Math 105 (2006), 35‐71), and can be applied to any type of grid (without “orthogonality” assumptions as for classical finite volume methods) and in any space dimension. We present numerical results on some irregular grids, and we prove, for both Stokes and Navier‐Stokes equations, the convergence of the scheme toward a solution of the continuous problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A posteriori error analysis of nonconforming finite volume elements for general second‐order elliptic PDEs

In this article, we study the a posteriori H¹ and L² error estimates for Crouzeix‐Raviart nonconforming finite volume element discretization of general second‐order elliptic problems in ?². The error estimators yield global upper and local lower bounds. Finally, numerical experiments are performed to illustrate the theoretical findings. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011