Local Maximum Principle Research Articles

Variational method for untangling and optimization of spatial meshes

A variational method that can provably construct 3D quasi-isometric mappings between domains of a complex shape is introduced. A local maximum principle for polyconvex mesh element distortion measures is formulated. It allows us to control the invertibility and distortion bounds for non-simplicial elements in the minimization process. A simple and efficient technique for construction of boundary orthogonal meshes suggested in Garanzha (2000) is applied to the construction of hexahedral meshes and thick prismatic mesh layers around complex shapes. The mesh untangling technique, which is a generalization of the penalty method suggested in Garanzha and Kaporin (1999), is verified on a wide set of challenging test problems. Another untangling technique based on theoretical ideas from Ivanenko (1997) is implemented and tested. It provably constructs admissible meshes using a finite number of minimization steps. A minimization technique for the mesh distortion functional is described. The approach is based on the global gradient search technique with preconditioning and domain decomposition for local mesh optimization and untangling. Application areas for explicit and implicit minimization methods are evaluated.

A maximum-principle preserving [formula omitted] finite element method for scalar conservation equations

This paper introduces a first-order viscosity method for the explicit approximation of scalar conservation equations with Lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension. Provided the lumped mass matrix is positive definite, the method is shown to satisfy the local maximum principle under a usual CFL condition. The method is independent of the cell type; for instance, the mesh can be a combination of tetrahedra, hexahedra, and prisms in three space dimensions.

Higher-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows

This paper deals with the multi-dimensional limiting process (MLP) for discontinuous Galerkin (DG) methods to compute compressible inviscid and viscous flows. The MLP, which has been quite successful in finite volume methods (FVM), is extended to DG methods for hyperbolic conservation laws. In previous works, the MLP was shown to possess several superior characteristics, such as the ability to control multi-dimensional oscillation efficiently and to capture both discontinuous and continuous multi-dimensional flow features accurately within the finite volume framework. In particular, the oscillation-control mechanism in multiple dimensions was established by combining the local maximum principle and the multi-dimensional limiting (MLP) condition, leading to the formulation of efficient and accurate MLP-u slope limiters.The MLP limiting strategy is now extended to the higher-order DG framework to develop the hierarchical MLP formulation on unstructured grids, which facilitates the capturing of compressible flow structures very accurately. By combining the behavior of local extrema with the augmented MLP condition, the hierarchical MLP method (the MLP-based troubled-cell markers and the MLP limiters in the RKDG formulation) is developed for arbitrary DG-Pn polynomial approximations. Through extensive numerical analyses and computations on unstructured grids, it is demonstrated that the proposed hierarchical DG-MLP method yields outstanding performance in resolving non-compressive as well as compressive flow features.

Higher order cell-based multidimensional upwind schemes for flow in porous media on unstructured grids

Standard reservoir simulation schemes employ single-point upstream weighting for convective flux approximation. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent.New locally conservative cell-based multi-dimensional upwind schemes and higher-order cell-based multi-dimensional upwind schemes that reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new higher-order schemes are comprised of two steps; (a) Higher-order approximation that corrects the directional diffusion of the approximation. (b) Truly multi-dimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along multi-dimensional flow paths. This approximation reduces cross-wind diffusion. Conditions on tracing direction and CFL number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations.Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve a range of unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

Infrared and visible image fusion technology based on directionlets transform

The article provides an infrared and visible image fusion algorithm based on directionlets transform. The registered original images were decomposed into the low-frequency and high-frequency coefficients by directionlets transform. Getting the mean of the low-frequency coefficients, applying the local variance maximum principle to the high-frequency coefficients, thereby the fusion coefficients of the fused image can be acquired. Finally, the fused image was obtained using inverse directionlets transform. The experiment indicates that the fusion algorithm can extract the original image features better. Compared with the traditional fusion algorithms, the fusion algorithm presented in this article provides better subjective visual effect, and the standard deviation and entropy value would be somewhat increased.

Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients

We establish local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic partial differential equations with unbounded ingredients.

Maximum principle for quasi-linear backward stochastic partial differential equations

In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDEs with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection–diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection–diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.

Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.

Quasi M-Matrix Multifamily Continuous Darcy-Flux Approximations with Full Pressure Support on Structured and Unstructured Grids in Three Dimensions

New multifamilies of flux-continuous vertex centered finite-volume methods are presented for the full-tensor pressure equation with general discontinuous coefficients for any cell type in three dimensions. The new schemes are flux continuous with full pressure support over each subcell with continuous pressure imposed across each control-volume subface, in contrast to earlier formulations. Full pressure continuity across subfaces leads to a quasi-positive formulation that minimizes spurious oscillations in discrete pressure solutions for strongly anisotropic full-tensor fields. The multifamily formulation permits maximum flexibility in quadrature, yielding improved solution resolution. The earlier methods are pointwise continuous in pressure and flux with tetrahedral pressure support, which leads to a more limited quadrature range, which is shown to cause spurious oscillations in the solution for strong full-tensor fields. An M-matrix analysis of the three-dimensional schemes identifies bounding limits for the schemes to possess a local discrete maximum principle. Conditions for the schemes to be positive definite are also derived.

A numerical method for mass conservative coupling between fluid flow and solute transport

We present a new coupled discretization approach for species transport in an incompressible fluid. The Navier–Stokes equations for the flow are discretized by the divergence-free Scott–Vogelius element on barycentrically refined meshes guaranteeing LBB stability. The convection–diffusion equation for species transport is discretized by the Voronoi finite volume method. In accordance to the continuous setting, due to the exact integration of the normal component of the flow through the Voronoi surfaces, the species concentration fulfills discrete global and local maximum principles. Besides of the numerical scheme itself, we present important aspects of its implementation. Further, for the case of homogeneous Dirichlet boundary conditions, we give a convergence proof for the coupled scheme. We report results of the application of the scheme to the interpretation of limiting current measurements in an electrochemical flow cell with cylindrical shape.

A family of MPFA finite-volume schemes with full pressure support for the general tensor pressure equation on cell-centered triangular grids

A new family of cell-centered finite-volume schemes is presented for solving the general full-tensor pressure equation of subsurface flow in porous media on arbitary unstructured triangulations. The new schemes are flux continuous and have full pressure support (FPS) over each subcell with continuous pressure imposed across each control-volume sub-interface, in contrast to earlier formulations. The earlier methods are point-wise continuous in pressure and flux with triangle-pressure-support (TPS) which leads to a more limited quadrature range. An M-matrix analysis identifies bounding limits for the schemes to posses a local discrete maximum principle. Conditions for the schemes to be positive definite are also derived. A range of computational examples are presented for unstructured triangular grids, including highly irregular grids, and the new FPS schemes are compared against the earlier pointwise continuous TPS formulations. The earlier pointwise TPS methods can induce strong spurious oscillations for problems involving strong full-tensor anisotropy where the M-matrix conditions are violated, and can lead to decoupled solutions in such cases. Unstructured cell-centered decoupling is investigated. In contrast to TPS, the new FPS formulation leads to well resolved solutions that are essentially free of spurious oscillations. A substantial degree of improved convergence behavior, for both pressure and velocity, is also observed in all convergence tests. This is particularly important for problems involving high anisotropy ratios. Also the new formulation proves to be highly beneficial for an upscaling example, where enhancement of convergence is highly significant for certain quadrature points, clearly demonstrating further advantages of the new formulation.

Some recent results on the Muskat problem

We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.

Higher Order Multidimensional Upwind Convection Schemes for Flow in Porous Media on Structured and Unstructured Quadrilateral Grids

Standard reservoir simulation schemes employ first order upwind schemes for approximation of the convective fluxes when multiple phases or components are present. These schemes introduce both coordinate-line numerical diffusion and cross-wind diffusion into the solution that is grid and geometry dependent. New locally conservative higher-order multidimensional upwind schemes that significantly reduce both directional and cross-wind diffusion are presented for convective flow approximation. The new schemes are composed of two steps; (a) higher order approximation that corrects the directional diffusion of the approximation and (b) truly multidimensional upwind approximation, which involves flux approximation using upwind information obtained by upstream tracing along wave-vector paths where wave information travels in multiple dimensions. This approximation reduces cross-wind diffusion. Conditions on the tracing direction and Courant–Friedrichs–Levy number lead to a local maximum principle that ensures stable solutions free of spurious oscillations. The schemes are coupled with full-tensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation, including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve structured and unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical cross-wind diffusion effects, leading to improved resolution of concentration and saturation fronts.

Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

AbstractHigher‐resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full‐tensor Darcy‐flux approximations. A sequence of non‐uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher‐order schemes are constructed using non‐uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full‐tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher‐order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal control of heterogeneous systems: Basic theory

A general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side-conditions. The heterogeneity is represented by a parameter taking values in an abstract measurable space, so that both continuous and discrete heterogeneity, as well as probabilistic heterogeneity without density, are included. A distributed and a lumped controls are involved, the latter appearing also in the side conditions. An existence theorem is proved for the uncontrolled system, and the sensitivity of the solution with respect to the control variables is estimated. The main result is an optimality condition in the form of the Pontryagin local maximum principle. A global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces. A number of possible applications are outlined: age-structured systems, size-structured systems, (nonlocal) advection–reaction equations, static parametric heterogeneity in epidemiology, and two-stage control systems with uncertain switching time.

Harnack Inequalities and ABP Estimates for Nonlinear Second‐Order Elliptic Equations in Unbounded Domains

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP‐type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville‐type theorem.

An algorithm for the construction of intrinsic delaunay triangulations with applications to digital geometry processing

The discrete Laplace–Beltrami operator plays a prominent role in many digital geometry processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate the use of the intrinsic Laplace–Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace–Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We detail an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace–Beltrami operator.

Polynomial convexity and Rossi's local maximum principle

Polynomial convexity and Rossi's local maximum principle

The repair paradigm: New algorithms and applications to compressible flow

The repair paradigm leads to several algorithms for redistributing mass, momentum and energy, while adhering to local maximum principles, as an adjunct to the remapping step in certain compressible flow codes that use remapping, such as Arbitrary–Lagrangian–Eulerian codes, or for just redistributing mass in advection codes. In the case of advection of a concentration, repair keeps the newly computed concentration in a cell between the maximum and minimum concentrations in neighboring old cells, thus guaranteeing at least that the new concentration is between zero and one. For compressible flow, density, velocity and internal energy are similarly constrained while maintaining conservation of mass, momentum and total energy. In this way, positive density and internal energy are achieved as a side effect. We propose a new algorithm, combining both local and global repair, that maintains causality and is efficient in a parallel computational setting. The local/global algorithm is independent of the order in which the distribution is performed, and it maintains 1D symmetry. This is applied to advection in two dimensions, and to, among others, the LeBlanc problem, the Sedov problem, and an interacting 2D blast wave problem. The latter is done with a Lagrangian code for which rezoning, remapping and repair are essential.