Discrete Maximum Principle Research Articles

MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS

In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

Discrete maximum principle for interior penalty discontinuous Galerkin methods

Abstract A class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of these conditions. The theoretical results are illustrated with numerical examples.

A parameter-free smoothness indicator for high-resolution finite element schemes

AbstractThis paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.

Overlapping Nonmatching Grid Method for the Ergodic Control Quasi Variational Inequalities

In this paper, we provide a maximum norm analysis of an overlapping Schwarz method on nonmatching grids for a quasi-variational inequalities related to ergodic control problems studied by M. Boulbrachene [1], where the “discount factor” (i.e., the zero order term) is set to 0, we use an overlapping Schwarz method on nonmatching grid which consists in decomposing the domain in two sub domains, where the discrete alternating Schwarz sequences in sub domains converge to the solution of the ergodic control IQV for the zero order term. For and under a discrete maximum principle we show that the discretization on each sub domain converges quasi-optimally in the norm to 0.

Analysis and Comparison of Different Approximations to Nonlocal Diffusion and Linear Peridynamic Equations

We consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models. Two classes of discretization methods are presented, including standard finite element methods and quadrature-based finite difference methods. We discuss the applicability of these approaches to nonlocal problems having various singular kernels and study basic numerical analysis issues. We illustrate the similarities and differences of the resulting nonlocal stiffness matrices and discuss whether discrete maximum principles can be established. We pay particular attention to the issue of convergence in both the nonlocal setting and the local limit. While it is known that the nonlocal models converge to corresponding differential equations in the local limit, we elucidate how such limiting behaviors may or may not be preserved in various discrete approximations. Our findings thus offer important insight into applications and simulations of nonlocal models.

An $$hp$$ -adaptive flux-corrected transport algorithm for continuous finite elements

This paper presents an $$hp$$ -adaptive flux-corrected transport algorithm for continuous finite elements. The proposed approach is based on a continuous Galerkin approximation with unconstrained higher-order elements in smooth regions and constrained $$P_1/Q_1$$ elements in the neighborhood of steep fronts. Smooth elements are found using a hierarchical smoothness indicator based on discontinuous higher-order reconstructions. A gradient-based error indicator determines the local mesh size $$h$$ and polynomial degree $$p$$ . The discrete maximum principle for linear/bilinear finite elements is enforced using a linearized flux-corrected transport (FCT) algorithm. The same limiting strategy is employed when it comes to constraining the $$L^2$$ projection of data from one finite-dimensional space into another. The new algorithm is implemented in the open-source software package Hermes. The use of hierarchical data structures that support arbitrary-level hanging nodes makes the extension of FCT to $$hp$$ -FEM relatively straightforward. The accuracy of the proposed method is illustrated by a numerical study for a two-dimensional benchmark problem with a known exact solution.

A positivity-preserving finite element method for chemotaxis problems in 3D

We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature.

Existence of solutions for a nonlinear algebraic system with a parameter

Taking advantage of a recent critical point theorem, the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established. Our goal was achieved requiring an appropriate behavior of the nonlinear term f, either at zero or at infinity, without symmetry conditions. In addition, for a suitable class of systems, a strong discrete maximum principle is presented.

A formally second‐order cell centred scheme for convection–diffusion equations on general grids

SUMMARYWe propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non‐conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second‐order spatial convergence rate for the Laplace equation on any unstructured two‐dimensional/three‐dimensional grids. As in the MUSCL method, the numerical convective fluxes are built with a prediction‐limitation process, which ensures that the discrete maximum principle is satisfied for pure convection problems. The limitation does not involve any geometrical reconstruction, thus allowing the use of completely general grids, in any space dimension. Copyright © 2012 John Wiley & Sons, Ltd.

Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

This paper extends the MOOD method proposed by the authors in [A high-order finite volume method for hyperbolic systems: Multi-Dimensional Optimal Order Detection (MOOD). J Comput Phys 2011;230:4028–50], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new detection criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the discrete maximum principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient.

Discrete maximum principles for nonlinear parabolic PDE systems

Discrete maximum principles for nonlinear parabolic PDE systems

Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems

SUMMARYIn this paper, two repair techniques are proposed for diamond schemes of anisotropic diffusion problems to ensure that the repaired solutions satisfy the discrete maximum principle. One of them is an extension of that in [Liska R, Shashkov M. Enforcing the discrete maximum principle for linear finite element solutions of second‐order elliptic problems. Communications in Computational Physics 2008; 3(4):852–877.] for linear finite element solutions, which is a local repair technique, and another is a new global repair technique. Both of them keep total energy conservation and are easy to be implemented in existing codes. Numerical examples show that these two repair techniques do not destroy the accuracy of solution for the diamond schemes on distorted meshes. Copyright © 2012 John Wiley & Sons, Ltd.

Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. Although our approach is presented for FEs, it admits natural extension to other numerical schemes, such as finite differences and finite volumes. For the postprocessing, a priori information about the monotonicity is assumed to be available, either for the whole domain or for a subdomain where the lost monotonicity is to be recovered. The obvious requirement is that such information is to be obtained without involving the exact solution, e.g. from expected symmetries of this solution.The postprocessing is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint that models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments.

Maximum Principles for $P1$-Conforming Finite Element Approximations of Quasi-linear Second Order Elliptic Equations

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial differential equations to finite element methods. The mathematical tools are based on the variational approach that was commonly used in the classical partial differential equation theory. The discrete maximum principles are established by assuming a property on the discrete variational form that is of global nature. In particular, the assumption on the variational form is verified when the finite element partition satisfies some angle conditions. For the general quasi-linear elliptic equation, these angle conditions indicate that each triangle or tetrahedron needs to be $\mathcal{O}(h^\alpha)$-acute in the sense that each angle $\alpha_{ij}$ (for the triangle) or interior dihedral angle $\alpha_{ij}$ (for the tetrahedron) must satisfy $\alpha_{ij}\le \pi/2-\gamma h^\alpha$ for some $\alpha\ge 0$ and $\gamma>0$. For the Poisson problem where the differential operator is given by the Laplacian, the angle requirement is the same as the existing ones: either all the triangles are nonobtuse or each interior edge is nonnegative. It should be pointed out that the analytical tools used in this paper are based on the powerful De Giorgi iterative method that has played important roles in the theory of partial differential equations. The mathematical analysis itself is of independent interest in the finite element analysis.

Minimal stencil finite volume scheme with the discrete maximum principle

We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.

Dynamic Optimization of a Polymer Flooding Process Based on Implicit Discrete Maximum Principle

Polymer flooding is one of the most important technologies for enhanced oil recovery (EOR). In this paper, an optimal control model of distributed parameter systems (DPSs) for polymer injection strategies is established, which involves the performance index as maximum of the profit, the governing equations as the fluid flow equations of polymer flooding, and some inequality constraints as polymer concentration and injection amount limitation. The optimal control model is discretized by full implicit finite‐difference method. To cope with the discrete optimal control problem (OCP), the necessary conditions for optimality are obtained through application of the calculus of variations and Pontryagin’s discrete maximum principle. A modified gradient method with new adjoint construction is proposed for the computation of optimal injection strategies. The numerical results of an example illustrate the effectiveness of the proposed method.

Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces

Solving a Helmholtz equation Δu + λu = f efficiently is a challenge for many applications. For example, the core part of many efficient solvers for the incompressible Navier-Stokes equations is to solve one or several Helmholtz equations. In this paper, two new finite difference methods are proposed for solving Helmholtz equations on irregular domains, or with interfaces. For Helmholtz equations on irregular domains, the accuracy of the numerical solution obtained using the existing augmented immersed interface method (AIIM) may deteriorate when the magnitude of λ is large. In our new method, we use a level set function to extend the source term and the PDE to a larger domain before we apply the AIIM. For Helmholtz equations with interfaces, a new maximum principle preserving finite difference method is developed. The new method still uses the standard five-point stencil with modifications of the finite difference scheme at irregular grid points. The resulting coefficient matrix of the linear system of finite difference equations satisfies the sign property of the discrete maximum principle and can be solved efficiently using a multigrid solver. The finite difference method is also extended to handle temporal discretized equations where the solution coefficient λ is inversely proportional to the mesh size.

The discrete maximum principle for Galerkin solutions of elliptic problems

Abstract This paper provides an equivalent characterization of the discrete maximum principle for Galerkin solutions of general linear elliptic problems. The characterization is formulated in terms of the discrete Green’s function and the elliptic projection of the boundary data. This general concept is applied to the analysis of the discrete maximum principle for the higher-order finite elements in one-dimension and to the lowest-order finite elements on simplices of arbitrary dimension. The paper surveys the state of the art in the field of the discrete maximum principle and provides new generalizations of several results.

Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes

This paper is concerned with the development of general-purpose algebraic flux correction schemes for continuous (linear and multilinear) finite elements. In order to enforce the discrete maximum principle (DMP), we modify the standard Galerkin discretization of a scalar transport equation by adding diffusive and antidiffusive fluxes. The result is a nonlinear algebraic system satisfying the DMP constraint. An estimate based on variational gradient recovery leads to a linearity-preserving limiter for the difference between the function values at two neighboring nodes. A fully multidimensional version of this scheme is obtained by taking the sum of local bounds and constraining the total flux. This new approach to algebraic flux correction provides a unified treatment of stationary and time-dependent problems. Moreover, the same algorithm is used to limit convective fluxes, anisotropic diffusion operators, and the antidiffusive part of the consistent mass matrix.The nonlinear algebraic system associated with the constrained Galerkin scheme is solved using fixed-point defect correction or a nonlinear SSOR method. A dramatic improvement of nonlinear convergence rates is achieved with the technique known as Anderson acceleration (or Anderson mixing). It blends a number of last iterates in a GMRES fashion, which results in a Broyden-like quasi-Newton update. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convection-dominated transport problems and anisotropic diffusion equations in 2D.

A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions

Convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level. In this paper, we are concerned with solving continuity equations that govern the evolution of volume fractions in Eulerian models of disperse two-phase flows. An implicit Galerkin finite element approximation is equipped with a flux limiter for the convective terms. The fully multidimensional limiting strategy is based on a flux-corrected transport (FCT) algorithm. This nonlinear high-resolution scheme satisfies a discrete maximum principle for divergence-free velocities and ensures positivity preservation for arbitrary velocity fields. To enforce an upper bound that corresponds to the maximum-packing limit, an FCT-like overshoot limiter is applied to the converged convective fluxes at the end of each time step. This postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model. Numerical results for 2D implosion problems illustrate the performance of the proposed limiting procedure.