A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

A $C^0$ -weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for $C^0$ functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete $H^2$ norm and the standard $H^1$ and $L^2$ norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order $(k+1-d)$ and the surface mass of order $(k+2-d)$ for the $P_{k+2}$ finite element functions in $d$ -dimensional space.     

Interpolation splines minimizing a semi-norm

Using S.L. Sobolev’s method, we construct the interpolation splines minimizing the semi-norm in $K_2(P_2)$ , where $K_2(P_2)$ is the space of functions $\phi $ such that $\phi ^{\prime } $ is absolutely continuous, $\phi ^{\prime \prime } $ belongs to $L_2(0,1)$ and $\int _0^1(\varphi ^{\prime \prime }(x)+\varphi (x))^2dx<\infty $ . Explicit formulas for coefficients of the interpolation splines are obtained. The resulting interpolation spline is exact for the trigonometric functions $\sin x$ and $\cos x$ . Finally, in a few numerical examples the qualities of the defined splines and $D^2$ -splines are compared. Furthermore, the relationship of the defined splines with an optimal quadrature formula is shown.     

Randomness Buys Depth for Approximate Counting

We show that the promise problem of distinguishing n-bit strings of relative Hamming weight \({1/2 + \Omega(1/{\rm lg}^{d-1} n)}\) from strings of weight \({1/2 - \Omega(1/{\rm \lg}^{d - 1} n)}\) can be solved by explicit, randomized (unbounded fan-in) poly(n)-size depth-d circuits with error \({\leq 1/3}\) , but cannot be solved by deterministic poly(n)-size depth-(d+1) circuits, for every \({d \geq 2}\) ; and the depth of both is tight. Our bounds match Ajtai’s simulation of randomized depth-d circuits by deterministic depth-(d + 2) circuits (Ann. Pure Appl. Logic; ’83) and provide an example where randomization buys resources. To rule out deterministic circuits, we combine Håstad’s switching lemma with an earlier depth-3 lower bound by the author (Computational Complexity 2009). To exhibit randomized circuits, we combine recent analyses by Amano (ICALP ’09) and Brody and Verbin (FOCS ’10) with derandomization. To make these circuits explicit, we construct a new, simple pseudorandom generator that fools tests \({A_1 \times A_2 \times \cdots \times A_{{\rm lg}{n}}}\) for \({A_i \subseteq [n], |A_{i}| = n/2}\) with error 1/n and seed length O(lg n), improving on the seed length \({\Omega({\rm lg}\, n\, {\rm lg}\, {\rm lg}\, n)}\) of previous constructions.     

On surface meshes induced by level set functions

The zero level set of a continuous piecewise-affine function with respect to a consistent tetrahedral subdivision of a domain in ${\mathbb {R}}^3$ is a piecewise-planar hyper-surface. We prove that if a family of consistent tetrahedral subdivions satisfies the minimum angle condition, then after a simple postprocessing this zero level set becomes a consistent surface triangulation which satisfies the maximum angle condition. We treat an application of this result to the numerical solution of PDEs posed on surfaces, using a $P_1$ finite element space on such a surface triangulation. For this finite element space we derive optimal interpolation error bounds. We prove that the diagonally scaled mass matrix is well-conditioned, uniformly with respect to $h$ . Furthermore, the issue of conditioning of the stiffness matrix is addressed.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler–Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to \((k+1)\) -degree Radau polynomials, when polynomials of total degree not exceeding \(k\) are used. These results allow us to prove that the \(k\) -degree LDG solution and its derivatives are \(\mathcal O (h^{k+3/2})\) superconvergent at the roots of \((k+1)\) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time \(t\) converge to the true errors at \(\mathcal O (h^{k+5/4})\) rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the \(L^2\) -norm converge to unity at \(\mathcal O (h^{1/2})\) rate. Our proofs are valid for arbitrary regular meshes and for \(P^k\) polynomials with \(k\ge 1\) , and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.     

Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach $h^{r+2}$ and even $h^{2r}$ , comparing with $h^{r+1}$ rate of the counterpart finite element method. Here $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.     

Local Discontinuous Galerkin Methods for One-Dimensional Second Order Fully Nonlinear Elliptic and Parabolic Equations

This paper is concerned with developing accurate and efficient nonstandard discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$ , two independent functions $q^-$ and $q^+$ are introduced to approximate one-sided derivatives of $u$ . Similarly, to capture the discontinuities of the second order derivative $u_{xx}$ , four independent functions $p^{- -}, p^{- +}, p^{+ -}$ , and $p^{+ +}$ are used to approximate one-sided derivatives of $q^-$ and $q^+$ . The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$ . An easy to verify set of criteria for constructing “good” numerical operators is also proposed. It consists of consistency and generalized monotonicity. To ensure such a generalized monotonicity property, the crux of the construction is to introduce the numerical moment in the numerical operator, which plays a critical role in the proposed LDG framework. The generalized monotonicity gives the LDG methods the ability to select the viscosity solution among all possible solutions. The proposed framework extends a companion finite difference framework developed by Feng and Lewis (J Comp Appl Math 254:81–98, 2013) and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes. Numerical experiments are also presented to demonstrate the accuracy, efficiency and utility of the proposed LDG methods.     

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send $\tilde{\varOmega}(m)$ messages and incur $\tilde{\varOmega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\tilde{\varOmega}(m)$ messages and $\tilde{\varOmega}(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using ${\tilde{O}}(\sqrt {n} + D)$ time) requires O(m+n ^3/2) messages, and the best known message-optimal algorithm (using ${\tilde{O}}(m)$ messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.     

Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fields

In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC⁰[p], the class of constant-depth circuits with unbounded fan-in MOD _p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ${\epsilon > 0}$ . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route of constructing our PRG, we prove two structural results for low-degree polynomials over finite fields that can be of independent interest.

1. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . Then, for every ${\epsilon > 0}$ , there exists a subset ${S \subset [n]}$ , whose size depends only on d and ${\epsilon}$ , such that ${\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}$ . Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.
2. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . If the distribution of f when applied to uniform zero–one bits is ${\epsilon}$ -far (in statistical distance) from its distribution when applied to biased bits, then for every ${\delta > 0}$ , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on ${\epsilon,\delta}$ and d) of lower degree polynomials.

     

On the power of unambiguity in log-space

We report progress on the NL versus UL problem.

We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including ${{{ReachFewL} \subseteq {UL}}}$ and ${{{LFew} \subseteq {UL}^{FewL}}}$

We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes.

We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that ${{{ULH} \subseteq {L}^{{promiseUL}}}}$ thus collapsing the UL hierarchy.

We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.

     

An efficient simulation algorithm on Kripke structures

A number of algorithms for computing the simulation preorder (and equivalence) on Kripke structures are available. Let $\varSigma $ denote the state space, ${\rightarrow }$ the transition relation and $P_{\mathrm {sim}}$ the partition of $\varSigma $ induced by simulation equivalence. While some algorithms are designed to reach the best space bounds, whose dominating additive term is $|P_{\mathrm {sim}}|^2$ , other algorithms are devised to attain the best time complexity $O(|P_{\mathrm {sim}}||{\rightarrow }|)$ . We present a novel simulation algorithm which is both space and time efficient: it runs in $O(|P_ {\mathrm {sim}}|^2 \log |P_{\mathrm {sim}}| + |\varSigma |\log |\varSigma |)$ space and $O(|P_{\mathrm {sim}}||{\rightarrow }|\log |\varSigma |)$ time. Our simulation algorithm thus reaches the best space bounds while closely approaching the best time complexity.     

A new PAC bound for intersection-closed concept classes

For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.     

Tradeoff lower lounds for stack machines

A space-bounded Stack Machine is a regular Turing Machine with a read-only input tape, several space-bounded read-write work tapes, and an unbounded stack. Stack Machines with a logarithmic space bound have been connected to other classical models of computation, such as polynomial-time Turing Machines (P) (Cook in J Assoc Comput Mach 18:4–18, 1971) and polynomial size, polylogarithmic depth, bounded fan-in circuits (NC) e.g., Borodin et al. (SIAM J Comput 18, 1989). In this paper, we present significant new lower bounds and techniques for Stack Machines. This comes in the form of a trade-off lower bound between space and number of passes over the input tape. Specifically, we give an explicit permuted inner product function such that any Stack Machine computing this function requires either ${\Omega (N^{1/4 - \epsilon})}$ space or ${\Omega (N^{1/4 - \epsilon})}$ number of passes for every constant ${\epsilon > 0}$ , where N is the input size. In the case of logarithmic space Stack Machines, this yields an unconditional ${\Omega (N^{1/4 - \epsilon})}$ lower bound for the number of passes. To put this result in perspective, we note that Stack Machines with logarithmic space and a single pass over the input can compute Parity, Majority, as well as certain languages outside NC. The latter follows from Allender (J Assoc Comput Mach 36:912–928, 1989), conditional on the widely believed complexity assumption that PSPACE ${\subsetneq}$ EXP. Our technique is a novel communication complexity reduction, thereby extending the already wide range of models of computation for which communication complexity can be used to obtain lower bounds. Informally, we show that a k-player number-in-hand (NIH) communication protocol for a base function f can efficiently simulate a space- and pass-bounded Stack Machine for a related function F, which consists of several “permuted” instances of f, bundled together by a combining function h. Trade-off lower bounds for Stack Machines then follow from known communication complexity lower bounds. The framework for this reduction was given by Beame & Huynh-Ngoc (2008), who used it to obtain similar trade-off lower bounds for Turing Machines with a constant number of pass-bounded external tapes. We also prove that the latter cannot efficiently simulate Stack Machines, conditional on the complexity assumption that E ${\not \subset}$ PSPACE. It is the treatment of an unbounded stack which constitutes the main technical novelty in our communication complexity reduction.     

Improved Approximation of Linear Threshold Functions

We prove two main results on how arbitrary linear threshold functions ${f(x) = {\rm sign}(w \cdot x - \theta)}$ over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is ${\epsilon}$ -close to a threshold function depending only on ${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$ many variables, where ${{\rm Inf}(f)}$ denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s well-known theorem (Friedgut in Combinatorica 18(1):474–483, 1998), which states that every Boolean function f is ${\epsilon}$ -close to a function depending only on ${2^{O({\rm Inf}(f)/\epsilon)}}$ many variables, for the case of threshold functions. We complement this upper bound by showing that ${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$ many variables are required for ${\epsilon}$ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is ${\epsilon}$ -close to a threshold function with integer weights at most ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$ This is an improvement, in the dependence on the error parameter ${\epsilon}$ , on an earlier result of Servedio (Comput Complex 16(2):180–209, 2007) which gave a ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180–209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.     

Leader election using loneliness detection

We consider the problem of leader election (LE) in single-hop radio networks with synchronized time slots for transmitting and receiving messages. We assume that the actual number n of processes is unknown, while the size u of the ID space is known, but is possibly much larger. We consider two types of collision detection: strong (SCD), whereby all processes detect collisions, and weak (WCD), whereby only non-transmitting processes detect collisions. We introduce loneliness detection (LD) as a key subproblem for solving LE in WCD systems. LD informs all processes whether the system contains exactly one process or more than one. We show that LD captures the difference in power between SCD and WCD, by providing an implementation of SCD over WCD and LD. We present two algorithms that solve deterministic and probabilistic LD in WCD systems with time costs of ${\mathcal{O}(\log \frac{u}{n})}$ and ${\mathcal{O}(\min( \log \frac{u}{n}, \frac{\log (1/\epsilon)}{n}))}$ , respectively, where ${\epsilon}$ is the error probability. We also provide matching lower bounds. Assuming LD is solved, we show that SCD systems can be emulated in WCD systems with factor-2 overhead in time. We present two algorithms that solve deterministic and probabilistic LE in SCD systems with time costs of ${\mathcal{O}(\log u)}$ and ${\mathcal{O}(\min ( \log u, \log \log n + \log (\frac{1}{\epsilon})))}$ , respectively, where ${\epsilon}$ is the error probability. We provide matching lower bounds.     

Arithmetizing Classes Around \textsf{NC} 1 and \textsf{L}

The parallel complexity class $\textsf{NC}$ ¹ has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, $\textsf{\#NC}$ ¹ and $\textsf{\#BWBP}$ . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in $\textsf{FLogDCFL}$ , while counting proof-trees in logarithmic width formulae has the same power as $\textsf{\#NC}$ ¹. We also consider polynomial-degree restrictions of $\textsf{SC}$ ⁱ , denoted $\textsf{sSC}$ ⁱ , and show that the Boolean class $\textsf{sSC}$ ¹ is sandwiched between $\textsf{NC}$ ¹ and $\textsf{L}$ , whereas $\textsf{sSC}$ ⁰ equals $\textsf{NC}$ ¹. On the other hand, the arithmetic class $\textsf{\#sSC}$ ⁰ contains $\textsf{\#BWBP}$ and is contained in $\textsf{FL}$ , and $\textsf{\#sSC}$ ¹ contains $\textsf{\#NC}$ ¹ and is in $\textsf{SC}$ ². We also investigate some closure properties of the newly defined arithmetic classes.     

A Unified Approach to Approximating Partial Covering Problems

An instance of the generalized partial cover problem consists of a ground set U and a family of subsets ${\mathcal{S}}\subseteq 2^{U}$ . Each element e??U is associated with a profit p(e), whereas each subset $S\in \mathcal{S}$ has a cost c(S). The objective is to find a minimum cost subcollection $\mathcal{S}'\subseteq \mathcal{S}$ such that the combined profit of the elements covered by $\mathcal{S}'$ is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e??U uncovered, we incur a penalty of ??(e). The goal is to identify a subcollection $\mathcal{S}'\subseteq \mathcal{S}$ that minimizes the cost of $\mathcal{S}'$ plus the penalties of uncovered elements. Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.     

On matrices, automata, and double counting in constraint programming

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables $\mathcal{M}$ , with the same constraint C defined by a finite-state automaton $\mathcal{A}$ on each row of $\mathcal{M}$ and a global cardinality constraint $\mathit{gcc}$ on each column of $\mathcal{M}$ . We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the $\mathit{gcc}$ constraints from the automaton $\mathcal{A}$ . The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We also provide a domain consistency filtering algorithm for the conjunction of lexicographic ordering constraints between adjacent rows of $\mathcal{M}$ and (possibly different) automaton constraints on the rows. We evaluate the impact of our methods in terms of runtime and search effort on a large set of nurse rostering problem instances.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the $L^2$ stability of the scheme and several optimal $L^2$ error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are $\mathcal O (h^{k+3/2})$ super close to particular projections of the exact solutions for $k$ th-degree polynomial spaces while computational results show higher $\mathcal O (h^{k+2})$ convergence rate. Our proofs are valid for arbitrary regular meshes and for $P^k$ polynomials with $k\ge 1$ , and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the $L^2$ -norm under mesh refinement.