Discrete mass conservation for porous media saturated flow

Global and local mass conservation for velocity fields associated with saturated porous media flow have long been recognized as integral components of any numerical scheme attempting to simulate these flows. In this work, we study finite element discretizations for saturated porous media flow that use Taylor–Hood (TH) and Scott–Vogelius (SV) finite elements. The governing equations are modified to include a stabilization term when using the TH elements, and we provide a theoretical result that shows convergence (with respect to the stabilization parameter) to pointwise mass‐conservative solutions. We also provide results using the SV approximation pair. These elements are pointwise divergence free, leading to optimal convergence rates and numerical solutions. We give numerical results to verify our theory and a comparison with standard mixed methods for saturated flow problems. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 625–640, 2014     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

List Colorings of K5‐Minor‐Free Graphs With Special List Assignments

The following question was raised by Bruce Richter. Let G be a planar, 3‐connected graph that is not a complete graph. Denoting by d(v) the degree of vertex v, is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v∈V(G)? More generally, we ask for which pairs (r, k) the following question has an affirmative answer. Let r and k be the integers and let G be a K₅‐minor‐free r‐connected graph that is not a Gallai tree (i.e. at least one block of G is neither a complete graph nor an odd cycle). Is G L‐list colorable for every list assignment L with |L(v)| = min{d(v), k} for all v∈V(G)? We investigate this question by considering the components of G[S_k], where S_k: = {v∈V(G)|d(v)8k} is the set of vertices with small degree in G. We are especially interested in the minimum distance d(S_k) in G between the components of G[S_k]. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:18–30, 2012     

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u ^k+1=P _Ω [u ^k −α _k d ^k ]. Interestingly, many of them can be paired such that [(u)\tilde]^k = P_\varOmega[u^k - b_kF(v^k)] = P_\varOmega[[(u)\tilde]^k - (d₂^k - G d₁^k)]\tilde{u}^{k} = P_{\varOmega}[u^{k} - \beta_{k}F(v^{k})] = P_{\varOmega}[\tilde {u}^{k} - (d_{2}^{k} - G d_{1}^{k})], where inf {β _k }>0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., d₁^kd_{1}^{k} and d₂^kd_{2}^{k} for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions (d₁^kd_{1}^{k} and d₂^kd_{2}^{k}) those studied proximal-like methods always utilize the unit step size namely α _k ≡1.     

Analogues of Horn’s theorem for finite unions of starshaped sets in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Fix k, d, 1 ≤ k ≤ d + 1. Let $ \mathcal{F} $ \mathcal{F} be a nonempty, finite family of closed sets in ℝ ^d , and let L be a (d − k + 1)-dimensional flat in ℝ ^d . The following results hold for the set T ≡ ∪{F: F in $ \mathcal{F} $ \mathcal{F} }. Assume that, for every k (not necessarily distinct) members F ₁, …, F _k of $ \mathcal{F} $ \mathcal{F} ,∪{F _i : 1 ≤ i ≤ k} is starshaped and the corresponding kernel contains a translate of L. Then T is starshaped, and its kernel also contains a translate of L.     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Critical Behavior in Almost Sure Central Limit Theory

Let X ₁,X ₂,… be i.i.d. random variables with EX ₁=0, EX ₁²=1 and let S _k =X ₁+⋅⋅⋅+X _k . We study the a.s. convergence of the weighted averages

Ubiquity of simplices in subsets of vector spaces over finite fields

We prove that a sufficiently large subset of the d-dimensional vector space over a finite field with q elements, $ \mathbb{F} $ _q ^d , contains a copy of every k-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.     

Geometric progressions in sumsets over finite fields

Given two sets A, B í \Bbb F_q^d{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d , the set of d dimensional vectors over the finite field \Bbb F_q{\Bbb F}_q with q elements, we show that the sumset A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\} contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector v ? \Bbb F_q^d{\bf v} \in {\Bbb F}_q^d and a nonsingular d × d matrix Λ whenever # A # B 3 20 q^2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k} . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

On the family of diophantine triples {<Emphasis Type="Italic">k</Emphasis> + 1, 4<Emphasis Type="Italic">k</Emphasis>, 9<Emphasis Type="Italic">k</Emphasis> + 3}

We prove that if k is a positive integer and d is a positive integer such that the product of any two distinct elements of the set {k + 1, 4k, 9k + 3, d} increased by 1 is a perfect square, then d = 144k ³ + 192k ² + 76k + 8.      

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Sufficient descent nonlinear conjugate gradient methods with conjugacy condition

In this paper, by the use of Gram-Schmidt orthogonalization, we propose a class of modified conjugate gradient methods. The methods are modifications of the well-known conjugate gradient methods including the PRP, the HS, the FR and the DY methods. A common property of the modified methods is that the direction generated by any member of the class satisfies g_k^Td_k=-||g_k||²g_{k}^{T}d_k=-\|g_k\|^2. Moreover, if line search is exact, the modified method reduces to the standard conjugate gradient method accordingly. In particular, we study the modified YT and YT+ methods. Under suitable conditions, we prove the global convergence of these two methods. Extensive numerical experiments show that the proposed methods are efficient for the test problems from the CUTE library.     

Local superconvergence of the derivative for tensor‐product block FEM

In this article, we shall discuss local superconvergence of the derivative for tensor‐product block finite elements over uniform partition for three‐dimensional Poisson's equation on the basis of Liu and Zhu (Numer Methods Partial Differential Eq 25 (2009) 999–1008). Assume that odd m ≥ 3, x₀ is an inner locally symmetric point of uniform rectangular partition \begin{align*}\mathcal{T}_{h}\end{align*} and ρ(x₀,?Ω) means the distance between x₀ and boundary ?Ω. Combining the symmetry technique (Wahlbin, Springer, 1995; Schatz, Sloan, and Wahlbin, SIAM J Numer Anal 33 (1996), 505–521; Schatz, Math Comput 67 (1998), 877–899) with weak estimates for tensor‐product block finite elements of degree m ≥ 3 [see Liu and Zhu, Numer Methods Partial Differential Eq 25 (2009) 999–1008] and the finite element theory of Green function in ??³ presented in this article, we propose the \begin{align*}O(h^{m+3}|\ln h|^{\frac{4}{3}}+h^{2m+2}|\ln h|^{\frac{4}{3}}\rho(x_{0},\partial\Omega)^{-m})\end{align*} convergence of the derivatives for tensor‐product block finite elements of degree m ≥ 3 on x₀. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 457–475, 2012     

Mesh‐centered finite differences from unconventional mixed‐hybrid nodal finite elements

It is shown how mesh‐centered finite differences can be obtained from unconventional mixed‐hybrid nodal finite elements. The classical Raviart‐Thomas schemes of index k (RTk) are based on interpolation parameters that are cell and/or edge moments. For the unconventional form (URTk), they become point values at Gaussian points. In particular, the scheme URT1 is fully described. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

On a class of partially ordered sets and their linear invariants

Let (, <) be a finite partially ordered set with rank function. Then is the disjoint union of the classes _k of elements of rank k and the order relation between elements in _k and _k+1 can be represented by a matrix S _k. We study partially ordered sets which satisfy linear recurrence relations of the type S _k (S _k ^T ) – c _k (S _{k – 1})^T S _{k – 1} = d _k ⁺ – c _k d _k ^– ) Id for all k and certain coefficients d _k ⁺, d _k ^- and c _k.     

Mixed finite element methods for incompressible flow: Stationary Stokes equations

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress‐velocity formulation. The pseudostress is approximated by the Raviart‐Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi‐optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Transcendence of the values of certain series with Hadamard's gaps

Transcendence of the number ?_k=0^￥ a^r_k \sum_{k=0}^\infty \alpha^{r_k} , where a \alpha is an algebraic number with 0 < | a | \mid\alpha\mid > 1 and {r_k}_k\geqq0 \{r_k\}_{k\geqq0} is a sequence of positive integers such that lim_k?￥ r_k+1/r_k = d ? \mathbbN \{1} \lim_{k\to\infty}\, r_{k+1}/r_k = d \in \mathbb{N}\, \backslash \{1\} , is proved by Mahler's method. This result implies the transcendence of the number ?_k=0^￥ a^kdk \sum_{k=0}^\infty \alpha^{kd^k} .