Scalar Diffusion Problems Research Articles

The Dual Approach to Optimal Control in the Coefficients of Nonlocal Nonlinear Diffusion

We derive the dual variational principle (principle of minimal complementary energy) for the nonlocal nonlinear scalar diffusion problem, which may be viewed as the nonlocal version of the p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p$$\\end{document}-Laplacian operator. We establish existence and uniqueness of solutions (two-point fluxes) as well as their quantitative stability, which holds uniformly with respect to the small parameter (nonlocal horizon) characterizing the nonlocality of the problem. We then focus on the nonlocal analogue of the classical optimal control in the coefficient problem associated with the dual variational principle, which may be interpreted as that of optimally distributing a limited amount of conductivity in order to minimize the complementary energy. We show that this nonlocal optimal control problem Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-converges to its local counterpart, when the nonlocal horizon vanishes.

Fast-reaction limit of reaction–diffusion systems with nonlinear diffusion

In this paper, we present an approach to characterizing fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction–diffusion equations, or one reaction–diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form [Formula: see text] in usual reaction–diffusion equation, which represent linear diffusion, by terms of form [Formula: see text], representing nonlinear diffusion. We prove the convergence in the fast-reaction limit [Formula: see text] that is determined by the unique solution of a certain scalar nonlinear diffusion problem.

Dynamics for nonlocal diffusion problems with a free boundary

In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain global existence, uniqueness and longtime behavior of solution of these two problems. The spreading-vanishing dichotomy is established, and some sufficient conditions for spreading and vanishing are also derived. For scalar models, we prove that accelerated spreading could happen if and only if a threshold condition is violated by kernel function. Then we discuss a classical Lotka-Volterra predator-prey model with nonlocal diffusions and a free boundary which can be seen as nonlocal diffusion counterpart of the model in the work of Wang (2014) [24].

A high‐order discontinuous Galerkin method for the incompressible Navier‐Stokes equations on arbitrary grids

SummaryA high‐order discontinuous Galerkin (DG) method is proposed in this work for solving the two‐dimensional steady and unsteady incompressible Navier‐Stokes (INS) equations written in conservative form on arbitrary grids. In order to construct the interface inviscid fluxes both in the continuity and in the momentum equations, an artificial compressibility term has been added to the continuity equation for relaxing the incompressibility constraint. Then, as the hyperbolic nature of the INS equations has been recovered, the local Lax‐Friedrichs (LLF) flux, which was previously developed in the context of hyperbolic conservation laws, is applied to discretize the inviscid term. Unlike the traditional artificial compressibility method, in this work, the artificial compressibility is introduced only for the construction of the inviscid numerical fluxes; therefore, a consistent discretization of the INS equations is obtained, irrespective of the amount of artificial compressibility used. What is more, as the LLF flux can be obtained directly and straightforward, no numerical iteration for solving an exact Riemann problem is entailed in our method. The viscous term is discretized by the direct DG method, which was developed based on the weak formulation of the scalar diffusion problems on structured grids. The performance and the accuracy of the method are demonstrated by computing a number of benchmark test cases, including both steady and unsteady incompressible flow problems. Due to its simplicity in implementation, our method provides an attractive alternative for solving the INS equations on arbitrary grids.

A direct discontinuous Galerkin method for the incompressible Navier–Stokes equations on arbitrary grids

A high-order direct discontinuous Galerkin (DG) method is proposed in this work for solving the two-dimensional steady and unsteady incompressible Navier–Stokes (INS) equations on arbitrary grids. The inviscid term is discretized by using a simplified artificial compressibility flux which can be obtained explicitly and straightforwardly without entailing any numerical iterative procedure for solving a Riemann problem of a nonlinear system. And the viscous term is discretized by the direct DG (DDG) method which was developed based on the weak formulation of the scalar diffusion problems on structured grids. A number of test cases are presented to assess the performance of the DDG method compared to the widely used BR2 method for solving the INS equations. The numerical results indicate that the DDG method can achieve the designed order of accuracy and is able to deliver comparable accuracy as the BR2 method. Due to its simplicity in implementation, the DDG method provides an attractive alternative for solving the INS equations on arbitrary grids.

Direct error in constitutive equation formulation for inverse heat conduction problem

SummaryWe present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.

Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with a so-called ideal interpolation that minimizes a weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap AMG setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the bootstrap AMG method with sparse interpolation matrix (and spanning multiple levels) converges faster than the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

A Symmetric Direct Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations

AbstractIn this work, we investigate the numerical approximation of the compressible Navier-Stokes equations under the framework of discontinuous Galerkin methods. For discretization of the viscous and heat fluxes, we extend and apply the symmetric direct discontinuous Galerkin (SDDG) method which is originally introduced for scalar diffusion problems. The original compressible Navier-Stokes equations are rewritten into an equivalent form via homogeneity tensors. Then, the numerical diffusive fluxes are constructed from the weak formulation of primal equations directly without converting the second-order equations to a first-order system. Additional numerical flux functions involving the jump of second order derivative of test functions are added to the original direct discontinuous Galerkin (DDG) discretization. A number of numerical tests are carried out to assess the practical performance of the SDDG method for the two dimensional compressible Navier-Stokes equations. These numerical results obtained demonstrate that the SDDG method can achieve the optimal order of accuracy. Especially, compared with the well-established symmetric interior penalty (SIP) method [18], the SDDG method can maintain the expected optimal order of convergence with a smaller penalty coefficient.

A direct discontinuous Galerkin method for the compressible Navier–Stokes equations on arbitrary grids

A Direct Discontinuous Galerkin (DDG) method is developed for solving the compressible Navier–Stokes equations on arbitrary grids in the framework of DG methods. The DDG method, originally introduced for scalar diffusion problems on structured grids, is extended to discretize viscous and heat fluxes in the Navier–Stokes equations. Two approaches of implementing the DDG method to compute numerical diffusive fluxes for the Navier–Stokes equations are presented: one is based on the conservative variables, and the other is based on the primitive variables. The importance of the characteristic cell size used in the DDG formulation on unstructured grids is examined. The numerical fluxes on the boundary by the DDG method are discussed. A number of test cases are presented to assess the performance of the DDG method for solving the compressible Navier–Stokes equations. Based on our numerical results, we observe that DDG method can achieve the designed order of accuracy and is able to deliver the same accuracy as the widely used BR2 method at a significantly reduced cost, clearly demonstrating that the DDG method provides an attractive alternative for solving the compressible Navier–Stokes equations on arbitrary grids owning to its simplicity in implementation and its efficiency in computational cost.

Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations

Abstract This paper presents two novel contributions on the recently introduced Mixed High-Order (MHO) methods [`Arbitrary order mixed methods for heterogeneous anisotropic diffusion on general meshes', preprint (2013)]. We first address the hybridization of the MHO method for a scalar diffusion problem and obtain the corresponding primal formulation. Based on the hybridized MHO method, we then design a novel, arbitrary order method for the Stokes problem on general meshes. A full convergence analysis is carried out showing that, when independent polynomials of degree k are used as unknowns (at elements for the pressure and at faces for each velocity component), the energy-norm of the velocity and the L 2-norm of the pressure converge with order (k + 1), while the L 2-norm of the velocity (super-)converges with order (k + 2). The latter property is not shared by other methods based on a similar choice of unknowns. The theoretical results are numerically validated in two space dimensions on both standard and polygonal meshes.

Sparsity in Bayesian inversion of parametric operator equations

We establish posterior sparsity in Bayesian inversion for systems governed by operator equations with distributed parameter uncertainty subject to noisy observation data δ. We generalize the results and algorithms introduced in C Schillings and C Schwab (2013 Inverse Problems 29 065011) for the particular case of scalar diffusion problems with random coefficients to broad classes of forward problems, including general elliptic and parabolic operators with uncertain coefficients, and in random domains. For countably parametric, deterministic representations of uncertain parameters in the forward problem, which belong to a specified sparsity class, we quantify analytic regularity of the likewise countably parametric, deterministic Bayesian posterior density with respect to a uniform prior on the uncertain parameter sequences and prove that the parametric, deterministic density of the Bayesian posterior belongs to the same sparsity class. Generalizing C Schillings and C Schwab (2013 Inverse Problems 29 065011) and C Schwab and A M Stuart (2012 Inverse Problems 28 045003) the forward problems are converted to countably parametric, deterministic operator equations. Computational Bayesian inversion amounts to numerically evaluating expectations of quantities of interest (QoIs) under the Bayesian posterior, conditional on noisy observation data. Our results imply, on the one hand, sparsity of Legendre (generalized) polynomial chaos expansions of the density of the Bayesian posterior with respect to uniform prior and, on the other hand, convergence rates for data-adaptive Smolyak integration algorithms for computational Bayesian estimation, which are independent of the dimension of the parameter space. We prove, mathematically and computationally, that for uncertain inputs with sufficient sparsity convergence rates are, in particular, superior to Markov chain Monte-Carlo sampling of the posterior, in terms of the number N of instances of the parametric forward problem to be solved.

Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.

The Elastoplast Discontinuous Galerkin (EDG) method for the Navier–Stokes equations

The present work details the Elastoplast (this name is a translation from the French “sparadrap”, a concept first applied by Yves Morchoisne for Spectral methods [1]) Discontinuous Galerkin (EDG) method to solve the compressible Navier–Stokes equations. This method was first presented in 2009 at the ICOSAHOM congress with some Cartesian grid applications. We focus here on unstructured grid applications for which the EDG method seems very attractive. As in the Recovery method presented by van Leer and Nomura in 2005 for diffusion, jumps across element boundaries are locally eliminated by recovering the solution on an overlapping cell. In the case of Recovery, this cell is the union of the two neighboring cells and the Galerkin basis is twice as large as the basis used for one element. In our proposed method the solution is rebuilt through an L 2 projection of the discontinuous interface solution on a small rectangular overlapping interface element, named Elastoplast, with an orthogonal basis of the same order as the one in the neighboring cells. Comparisons on 1D and 2D scalar diffusion problems in terms of accuracy and stability with other viscous DG schemes are first given. Then, 2D results on acoustic problems, vortex problems and boundary layer problems both on Cartesian or unstructured triangular grids illustrate stability, precision and versatility of this method.

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

We consider sparse linear systems, where a set of “interior” degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the “interior” principal submatrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach and show how the elimination process can exploit a particular “subzonal” discretization to maintain the sparsity of the Schur complement. We also investigate algebraic multigrid (AMG) solution algorithms applied to the reduced problem, and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge–Stüben AMG, applied to scalar diffusion problems [V. Henson and U. Yang, Appl. Numer. Math., 41 (2002), pp. 155–177], and the auxiliary-space Maxwell solver (AMS) for electromagnetic diffusion applications [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604–623]. In the electromagnetic case, we establish algebraically a reduced version of the Hiptmair–Xu decomposition from [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509] and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of two-dimensional, three-dimensional, and axisymmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time.

Improving algebraic multigrid interpolation operators for linear elasticity problems

AbstractLinear systems arising from discretizations of systems of partial differential equations can be challenging for algebraic multigrid (AMG), as the design of AMG relies on assumptions based on the near‐nullspace properties of scalar diffusion problems. For elasticity applications, the near‐nullspace of the operator includes the so‐called rigid body modes (RBMs), which are not adequately represented by the classical AMG interpolation operators. In this paper we investigate several approaches for improving AMG convergence on linear elasticity problems by explicitly incorporating the near‐nullspace modes in the range of the interpolation. In particular, we propose two new methods for extending any initial AMG interpolation operator to exactly fit the RBMs based on the introduction of additional coarse degrees of freedom at each node. Though the methodology is general and can be used to fit any set of near‐nullspace vectors, we focus on the RBMs of linear elasticity in this paper. The new methods can be incorporated easily into existing AMG codes, do not require matrix inversions, and do not assume an aggregation approach or a finite element framework. We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems. Copyright © 2009 John Wiley & Sons, Ltd.

Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data

We introduce a stochastic Galerkin mixed formulation of the steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use with the minimum residual method. We derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters.

Robust norm equivalencies for diffusion problems

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem. In this paper we present such a robust norm equivalence for the model problem $- \nabla \omega \nabla u=f$ with a scalar diffusion coefficient $\omega$ in $\Omega \subset \mathbb {R}^2$. Our estimates involve only very weak information about $\omega$, and the results are applicable for a large class of diffusion coefficients. Namely, we require $\omega$ to be in the Muckenhoupt class $A_{1}(\Omega )$, a function class well-studied in harmonic analysis. The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.

3D multidomain BEM for solving the Laplace equation

An efficient 3D multidomain boundary element method (BEM) for solving problems governed by the Laplace equation is presented. Integral boundary equations are discretized using mixed boundary elements. The field function is interpolated using a continuous linear function while its derivative in a normal direction is interpolated using a discontinuous constant function over surface boundary elements. Using a multidomain approach, also known as the subdomain technique, sparse system matrices similar to the finite element method (FEM) are obtained. Interface boundary conditions between subdomains leads to an over-determined system matrix, which is solved using a fast iterative linear least square solver. The accuracy and robustness of the developed numerical algorithm is presented on a scalar diffusion problem using simple cube geometry and various types of meshes. Efficiency is demonstrated with potential flow around the complex geometry of a fighter airplane using tetrahedral mesh with over 100,000 subdomains on a personal computer.

Two Simple Derivations of Universal Bounds for the C.B.S. Inequality Constant

Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations.

Simplified methods and a posteriori error estimation for the homogenization of representative volume elements (RVE)

Homogenization techniques usually rely on solving a boundary value problem on the representative element volume (RVE). This problem is generally complex to solve when the micro-structure is realistic, especially in three dimensions. In this paper, we develop two simplified methods providing approximate micro-fields over the RVE. These fields yield upper and lower bounds to the exact homogenized property. The a posteriori estimation of the modeling error introduced by the simplified methods is thus straightforward. Both simplified methods are based on a two-scale strategy. The RVE is decomposed into subdomains over which the solution is sought as a smooth part (meso-scale) plus a correction (micro-scale). The correction is expressed in terms of smooth part through a prolongation operator. This operation is performed independently on each subdomain and is thus readily parallelizable. Then, the smooth part of the solution is obtained by solving a ‘meso’ problem involving all the subdomains. In the numerical experiments, we consider 2-D linear scalar diffusion problems with periodic boundary conditions on the RVE. The RVE is made of a two-phase material consisting of a matrix in which circular or elliptical inclusions are distributed randomly. Numerical examples are computed in a parallel computation done on a cluster of 16 Intel P.C.s.