Discrete maximum principles for FE solutions of nonstationary diffusion‐reaction problems with mixed boundary conditions

In this article, we derive and discuss sufficient conditions for providing validity of the discrete maximum principle for nonstationary diffusion‐reaction problems with mixed boundary conditions, solved by means of simplicial finite elements and the θ time discretization method. The theoretical analysis is supported by numerical experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Discrete maximum principle for higher-order finite elements in 1D

We formulate a sufficient condition on the mesh under which we prove the discrete maximum principle (DMP) for the one-dimensional Poisson equation with Dirichlet boundary conditions discretized by the -FEM. The DMP holds if a relative length of every element in the mesh is bounded by a value , where is the polynomial degree of the element . The values are calculated for .

     

Maximum principle for the finite element solution of time‐dependent anisotropic diffusion problems

Preservation of the maximum principle is studied for the combination of the linear finite element method in space and the θ ‐method in time for solving time‐dependent anisotropic diffusion problems. It is shown that the numerical solution satisfies a discrete maximum principle when all element angles of the mesh measured in the metric specified by the inverse of the diffusion matrix are nonobtuse, and the time step size is bounded below and above by bounds proportional essentially to the square of the maximal element diameter. The lower bound requirement can be removed when a lumped mass matrix is used. In two dimensions, the mesh and time step conditions can be replaced by weaker Delaunay‐type conditions. Numerical results are presented to verify the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Failure of the discrete maximum principle for an elliptic finite element problem

There has been a long-standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green's function is positive for triangular meshes allowing sufficiently good approximation of functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green's functions that are of independent interest.

     

On the existence of maximum principles in parabolic finite element equations

In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem. We then show that for the lumped mass method the above conditions on the triangulation are essentially sharp. This is in contrast to the elliptic case in which the requirements are weaker. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction or a positive operator.

     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

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     

Nonsmooth maximum principle for infinite-horizon problems

In this paper, we consider a class of infinite-horizon discounted optimal control problems with nonsmooth problem data. A maximum principle in terms of differential inclusions with a Michel type transversality condition is given. It is shown that, when the discount rate is sufficiently large, the problem admits normal multipliers and a strong transversality condition holds. A relationship between dynamic programming and the maximum principle is also given.The author is indebted to Francis Clarke for helpful suggestions and discussions.     

Patch‐wise local projection stabilized finite element methods for convection–diffusion–reaction problems

In this article, we develop patch‐wise local projection‐stabilized conforming and nonconforming finite element methods for the convection–diffusion–reaction problems. It is a composition of the standard Galerkin finite element method, the patch‐wise local projection stabilization, and weakly imposed Dirichlet boundary conditions on the discrete solution. In this paper, a priori error analysis is established with respect to a patch‐wise local projection norm for the conforming and the nonconforming finite element methods. The numerical experiments confirm the efficiency of the proposed stabilization technique and validate the theoretical convergence rates.     

On the sufficiency of the linear maximum principle for discrete-time control problems

This note presents a family of linear maximum principles for the discrete-time optimal control problem, derived from the saddle-point theorem of mathematical programming. Some simple examples illustrate the applicability of the main theoretical results.     

On the maximum principle for linear parabolic equations

We prove extensions of our previous estimates for linear elliptic equations with inhomogeneous terms in L ^p spaces, p ≤ n to linear parabolic equations with inhomogeneous terms in L ^p , p ≤ n + 1. As with the elliptic case, our results depend on restrictions on parabolicity determined by certain subcones of the positive cone . They also extend the maximum principle of Krylov for the case p = n + 1, corresponding to the usual parabolicity.     

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.     

On the strong maximum principle for degenerate parabolic equations

Variants of the strong maximum principle are established for subsolutions to degenerate parabolic equations for which the standard version of the strong maximum principle does not hold. The results are formulated for viscosity solutions.     

A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On the refined maximum principle for degenerate elliptic and parabolic problems

We extend the refined maximum principle in [H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47 (1994) 47–92] to degenerate elliptic and parabolic equations with unbounded coefficients. Then we discuss the well-posedness of the corresponding Dirichlet boundary value problems.     

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

A nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order in the mesh size for the L²‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Pointwise maximum principle for convex optimal control problems with mixed control-phase variable inequality constraints

The validity of a global pointwise maximum principle is proved for a class of convex optimal control problems with mixed control-phase variable inequality constraints. No compatibility hypotheses are required, and singular multipliers are avoided.     

On the maximum principle for deterministic impulse control problems

This paper is devoted to a simple and direct proof of a version of the Blaquiere's maximum principle for deterministic impulse control problems.     

Mixed finite element approximation of eddy current problems

Finite element approximations of eddy current problems thatare entirely based on the magnetic field H are haunted by theneed to enforce the algebraic constraint curl H=0 in non-conductingregions. As an alternative to techniques employing combinatorialSeifert (cutting) surfaces, in order to introduce a scalar magneticpotential we propose mixed multi-field formulations, which enforcethe constraint in the variational formulation. In light of thefact that the computation of cutting surfaces is expensive,the mixed finite element approximation is a viable option despitethe increased number of unknowns.