Existence of solution of a finite volume scheme preserving maximum principle for diffusion equations

In this article, a cell‐centered finite volume scheme preserving maximum principle for diffusion equations with scalar coefficients is developed. The construction of the scheme consists of three steps: at first the discrete normal flux is obtained by a linear combination of two single‐sided fluxes, then the tangential term of the normal flux is modified by using a nonlinear combination of two single‐sided tangential fluxes, finally the auxiliary unknowns in the tangential fluxes are calculated by the convex combinations of the cell‐centered unknowns. It is proved that this nonlinear scheme satisfies the discrete maximum principle (DMP). Moreover, the existence of a solution of the nonlinear scheme is proved by using the Brouwer's fixed point theorem and the bounded estimates. Numerical experiments are presented to show that the scheme not only satisfies DMP, but also obtains the second‐order accuracy and conservation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 80–96, 2018     

Implicit finite element schemes for stationary compressible particle-laden gas flows

The derivation of macroscopic models for particle-laden gas flows is reviewed. Semi-implicit and Newton-like finite element methods are developed for the stationary two-fluid model governing compressible particle-laden gas flows. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers. This is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters and the additional strong nonlinearity caused by interfacial coupling terms. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. The original Jacobian is replaced by a low-order approximation. Special emphasis is laid on the numerical treatment of weakly imposed boundary conditions. It is shown that the proposed approach offers unconditional stability and faster convergence rates for increasing CFL numbers. The strongly coupled solver is compared to operator splitting techniques, which are shown to be less robust.     

A vertex‐centered and positivity‐preserving scheme for anisotropic diffusion equations on general polyhedral meshes

We propose a new nonlinear positivity‐preserving finite volume scheme for anisotropic diffusion problems on general polyhedral meshes with possibly nonplanar faces. The scheme is a vertex‐centered one where the edge‐centered, face‐centered, and cell‐centered unknowns are treated as auxiliary ones that can be computed by simple second‐order and positivity‐preserving interpolation algorithms. Different from most existing positivity‐preserving schemes, the presented scheme is based on a special nonlinear two‐point flux approximation that has a fixed stencil and does not require the convex decomposition of the co‐normal. More interesting is that the flux discretization is actually performed on a fixed tetrahedral subcell of the primary cell, which makes the scheme very easy to be implemented on polyhedral meshes with star‐shaped cells. Moreover, it is suitable for polyhedral meshes with nonplanar faces, and it does not suffer the so‐called numerical heat‐barrier issue. The truncation error is analyzed rigorously, while the Picard method and its Anderson acceleration are used for the solution of the resulting nonlinear system. Numerical experiments are also provided to demonstrate the second‐order accuracy and well positivity of the numerical solution for heterogeneous and anisotropic diffusion problems on severely distorted grids.     

Construction of locally conservative fluxes for the SUPG method

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015     

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.     

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.     

A comparative Fourier analysis of discontinuous Galerkin schemes for advection–diffusion with respect to BR1, BR2, and local discontinuous Galerkin diffusion discretization

This work compares the wave propagation properties of discontinuous Galerkin (DG) schemes for advection–diffusion problems with respect to the behavior of classical discretizations of the diffusion terms, that is, two versions of the local discontinuous Galerkin (LDG) scheme as well as the BR1 and the BR2 scheme. The analysis highlights a significant difference between the two possible ways to choose the alternating LDG fluxes showing that the variant that is inconsistent with the upwind advective flux is more accurate in case of advection–diffusion discretizations. Furthermore, whereas for the BR1 scheme used within a third order DG scheme on Gauss-Legendre nodes, a higher accuracy for well-resolved problems has previously been observed in the literature, this work shows that higher accuracy of the BR1 discretization only holds for odd orders of the DG scheme. In addition, this higher accuracy is generally lost on Gauss–Legendre–Lobatto nodes.     

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5- and 7-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme.     

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.     

A cell edge‐based linearity‐preserving scheme for diffusion problems on two‐dimensional unstructured grids

We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell‐centered primary unknowns and cell edge‐based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell‐centered one. The derivation of the scheme satisfies the linearity‐preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge‐based linearity‐preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd.     

Finite element simulation of compressible particle-laden gas flows

A macroscopic two-fluid model of compressible particle-laden gas flows is considered. The governing equations are discretized by a high-resolution finite element method based on algebraic flux correction. A multidimensional limiter of TVD type is employed to constrain the local characteristic variables for the continuous gas phase and conservative fluxes for a suspension of solid particles. Special emphasis is laid on the efficient computation of steady state solutions at arbitrary Mach numbers. To avoid stability restrictions and convergence problems, the characteristic boundary conditions are imposed weakly and treated in a fully implicit manner. A two-way coupling via the interphase drag force is implemented using operator splitting. The Douglas-Rachford scheme is found to provide a robust treatment of the interphase exchange terms within the framework of a fractional-step solution strategy. Two-dimensional simulation results are presented for a moving shock wave and for a steady nozzle flow.     

Nonlinear Galerkin method and two-step method for the Navier-Stokes equations

This article represents a new nonlinear Galerkin scheme for the Navier-Stokes equations. This scheme consists of a nonlinear Galerkin finite element method and a two-step difference method. Moreover, we also provide a Galerkin scheme. By convergence analysis, two numerical schemes have the same second-order convergence accuracy for the spatial discretization and time discretization if H is chosen such that H = O(h^2/3). However, the nonlinear Galerkin scheme is simpler than the Galerkin scheme, namely, this scheme can save a large amount of computational time. © 1996 John Wiley & Sons, Inc.     

Monotone combined edge finite volume–finite element scheme for Anisotropic Keller–Segel model

In this article, a new numerical scheme for a degenerate Keller–Segel model with heterogeneous anisotropic tensors is treated. It is well‐known that standard finite volume scheme not permit to handle anisotropic diffusion without any restrictions on meshes. Therefore, a combined finite volume‐nonconforming finite element scheme is introduced, developed, and studied. The unknowns of this scheme are the values at the center of cell edges. Convergence of the approximate solution to the continuous solution is proved only supposing the shape regularity condition for the primal mesh. This scheme ensures the validity of the discrete maximum principle under the classical condition that all transmissibilities coefficients are positive. Therefore, a nonlinear technique is presented, as a correction of the diffusive flux, to provide a monotone scheme for general tensors. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1030–1065, 2014     

Galerkin and inertial manifold methods for finite-dimensional approximation of nonlocal equations of nonlinear optics

The article examines the class of models of nonlinear optical systems with two-dimensional feedback described by a nonlocal diffusion equation and a Schrödinger equation. Rate of convergence bounds are obtained for the Galerkin scheme with an eigenelement basis. The existence of a finite-dimensional inertial manifold with the global attraction property is established.     

Optimal scaling parameters for sparse grid discretizations

We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close‐to‐optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters that result in improved asymptotic convergence properties, and finally, we use the OptiCom method as a variable nonlinear preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution of double nonlinear problems in porous media by a combined finite volume–finite element algorithm

The combined finite volume–finite element scheme for a double nonlinear parabolic convection-dominated diffusion equation which models the variably saturated flow and contaminant transport problems in porous media is extended. Whereas the convection is approximated by a finite volume method (Multi-Point Flux Approximation), the diffusion is approximated by a finite element method. The scheme is fully implicit and involves a relaxation-regularized algorithm. Due to monotonicity and conservation properties of the approximated scheme and in view of the compactness theorem we show the convergence of the numerical scheme to the weak solution. Our scheme is applied for computing two dimensional examples with different degrees of complexity. The numerical results demonstrate that the proposed scheme gives good performance in convergence and accuracy.     

THE UPWIND FINITE ELEMENT SCHEME AND MAXIMUM PRINCIPLE FOR NONLINEAR CONVECTION-DIFFUSION PROBLEM

In this paper, a kind of partial upwind finite element scheme is studied for twodimensional nonlinear convection-diffusion problem. Nonlinear convection term approximated by partial upwind finite element method considered over a mesh dual to the triangular grid, whereas the nonlinear diffusion term approximated by Galerkin method. A linearized partial upwind finite element scheme and a higher order accuracy scheme are constructed respectively. It is shown that the numerical solutions of these schemes preserve discrete maximum principle. The convergence and error estimate are also given for both schemes under some assumptions. The numerical results show that these partial upwind finite element scheme are feasible and accurate.     

A control volume technique based on integrated RBFNs for the convection–diffusion equation

This article reports a new high‐order control‐volume discretization for the convection–diffusion equation in one and two dimensions. Diffusive fluxes at the faces of a control volume and other terms embracing the unknown field variable are all approximated using one‐dimensional integrated radial‐basis‐function networks; line integrals involving these fluxes and other integrals are evaluated using a high‐order numerical integration scheme. The accuracy of the proposed technique is investigated numerically through the solution of several linear and nonlinear test problems, including a benchmark thermally driven cavity flow. High‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Numerical scheme for a scalar conservation law with memory

We consider approximation of solutions to conservation laws with memory represented by a Volterra term with a smooth decreasing but possibly unbounded kernel. The numerical scheme combines Godunov method with a treatment of the integral term following from product integration rules. We prove stability for both linear and nonlinear flux functions and demonstrate the expected order of convergence using numerical experiments. The problem is motivated by modeling advective transport in heterogeneous media with subscale diffusion.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 239–264, 2014