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.     

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.     

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.     

Superconvergence of Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

In this paper, we present a unified approach to study superconvergence behavior of the local discontinuous Galerkin (LDG) method for high-order time-dependent partial differential equations. We select the third and fourth order equations as our models to demonstrate this approach and the main idea. Superconvergence results for the solution itself and the auxiliary variables are established. To be more precise, we first prove that, for any polynomial of degree k, the errors of numerical fluxes at nodes and for the cell averages are superconvergent under some suitable initial discretization, with an order of \(O(h^{2k+1})\). We then prove that the LDG solution is \((k+2)\)-th order superconvergent towards a particular projection of the exact solution and the auxiliary variables. As byproducts, we obtain a \((k+1)\)-th and \((k+2)\)-th order superconvergence rate for the derivative and function value approximation separately at a class of Radau points. Moreover, for the auxiliary variables, we, for the first time, prove that the convergence rate of the derivative error at the interior Radau points can reach as high as \(k+2\). Numerical experiments demonstrate that most of our error estimates are optimal, i.e., the error bounds are sharp.     

Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE’s by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of $2$ -coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger $2$ -coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).     

Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws

In this paper, we establish negative-order norm estimates for the accuracy of discontinuous Galerkin (DG) approximations to scalar nonlinear hyperbolic equations with smooth solutions. For these special solutions, we are able to extract this “hidden accuracy” through the use of a convolution kernel that is composed of a linear combination of B-splines. Previous investigations into extracting the superconvergence of DG methods using a convolution kernel have focused on linear hyperbolic equations. However, we now demonstrate that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter for scalar nonlinear hyperbolic equations. Furthermore, we provide theoretical error estimates for the DG solutions that show improvement to $(2k+m)$ -th order in the negative-order norm, where $m$ depends upon the chosen flux.     

The Compact Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

A compact discontinuous Galerkin method (CDG) is devised for nearly incompressible linear elasticity, through replacing the global lifting operator for determining the numerical trace of stress tensor in a local discontinuous Galerkin method (cf. Chen et al., Math Probl Eng 20, 2010) by the local lifting operator and removing some jumping terms. It possesses the compact stencil, that means the degrees of freedom in one element are only connected to those in the immediate neighboring elements. Optimal error estimates in broken energy norm, $H^1$ -norm and $L^2$ -norm are derived for the method, which are uniform with respect to the Lamé constant $\lambda .$ Furthermore, we obtain a post-processed $H(\text{ div})$ -conforming displacement by projecting the displacement and corresponding trace of the CDG method into the Raviart–Thomas element space, and obtain optimal error estimates of this numerical solution in $H(\text{ div})$ -seminorm and $L^2$ -norm, which are uniform with respect to $\lambda .$ A series of numerical results are offered to illustrate the numerical performance of our method.     

Stability of Galerkin multistep procedures in time-homogeneous hyperbolic problems

Linear and time-homogeneous hyperbolic initial boundary value problems are approximated using Galerkin procedures for the space directions and linear multistep methods for the time direction. At first error bounds are proved for multistep methods having a stability interval [?ω, 0], 0<ω, and systemsY″=AY+C(t) under the condition that \(\Delta t^2 \left\| A \right\| \leqslant \omega \) Δt time step. Then these error bounds are applied to derive bounds for the error in hyperbolic problems. The result shows that the initial error and the discretization error grow liket andt ² respectively. But the initial error is multiplied with a factor which becomes large if the mesh width of the space discretization is small.     

A Coupling of Local Discontinuous Galerkin and Natural Boundary Element Method for Exterior Problems

In this paper, we apply the coupling of local discontinuous Galerkin (LDG) and natural boundary element method(NBEM) to solve a two-dimensional exterior problem. As a consequence, the main features of LDG and NBEM are maintained and hence the coupled approach benefits from the advantages of both methods. Referring to Gatica et al. (Math. Comput. 79(271):1369?C1394, 2010), we employ LDG subspaces whose functions are continuous on the coupling boundary. In this way, the primitive variables become the only boundary unknown, and hence the total number of unknown functions is reduced. We prove the stability of the new discrete scheme and derive an a priori error estimate in the energy norm. Some numerical examples conforming the theoretical results are provided.     

Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations

In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn–Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods for the CH equation will require severe time step restriction $(\varDelta t \sim O(\varDelta x^4))$ , so implicit methods are used to remove time step restriction. Implicit methods will result in large system of algebraic equations and a fast solver is essential. The multigrid (MG) method is used to solve the algebraic equations efficiently. The Local Mode Analysis method is used to analyze the convergence behavior of the linear MG method. The discrete energy stability for the CH equations with a special homogeneous free energy density $\Psi (u)=\frac{1}{4}(1-u^2)^2$ is proved based on the convex splitting method. We show that the number of iterations is independent of the problem size. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. We numerically show the optimal complexity of the MG solver for $\mathcal{P }^1$ element. For $\mathcal{P }^2$ approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.     

Stability Analysis and Error Estimates of Semi-implicit Spectral Deferred Correction Coupled with Local Discontinuous Galerkin Method for Linear Convection–Diffusion Equations

In this paper, we focus on the theoretical analysis of the second and third order semi-implicit spectral deferred correction (SDC) time discretization with local discontinuous Galerkin (LDG) spatial discretization for the one-dimensional linear convection–diffusion equations. We mainly study the stability and error estimates of the corresponding fully discrete scheme. Based on the Picard integral equation, the SDC method is driven iteratively by either the explicit Euler method or the implicit Euler method. It is easy to implement for arbitrary order of accuracy. For the semi-implicit SDC scheme, the iteration and the left-most endpoint involved in the integral for the implicit part increase the difficulty of the theoretical analysis. To be more precise, the test functions are more complex and the energy equations are more difficult to construct, compared with the Runge–Kutta type semi-implicit schemes. Applying the energy techniques, we obtain both the second and third order semi-implicit SDC time discretization with LDG spatial discretization are stable provided the time step \(\tau \le \tau _{0}\), where the positive \(\tau _{0}\) depends on the diffusion and convection coefficients and is independent of the mesh size h. We then obtain the optimal error estimates for the corresponding fully discrete scheme under the condition \(\tau \le \tau _{0}\) with similar technique for stability analysis. Numerical examples are presented to illustrate our theoretical results.     

Decomposition of realizable fuzzy relations

This paper deals with the decomposition problem of realizable fuzzy relations. First, given a realizable fuzzy relation $A$ , a method to construct a fuzzy relation $B$ such that $A=B\odot B^T$ (where $\odot $ is the max–min composition, $B^T$ denotes the transpose of $B$ ) is proposed. Then it is proved that the content of a realizable fuzzy relation is equal to the chromatic number of a simple graph generated by the realizable fuzzy relation. Therefore, many existing algorithms (including exact and heuristic algorithms) developed to find the chromatic number or to get an upper bound on chromatic number of a graph can be applied to solve the calculating problem of the content of a realizable fuzzy relation.     

Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations

Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like \(h^{p+4}\exp (h^2Lt)\) , where \(p\) is the order of the method, and \(L\) depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.     

H-adaptive Mesh Method with Double Tolerance Adaptive Strategy for Hyperbolic Conservation Laws

The numerical method used to solve hyperbolic conservation laws is often an explicit scheme. As a commonly used technique to improve the quality of numerical simulation, the $h$ -adaptive mesh method is adopted to resolve sharp structures in the solution. Since the computational costs of altering the mesh and solving the PDEs are comparable, too often the mesh adaption triggered may bring down the overall efficiency of solving hyperbolic conservation laws using $h$ -adaptive mesh method. In this paper, we propose a so-called double tolerance adaptive strategy to optimize the overall numerical efficiency by reducing the number of mesh adaptions, as well as preserving the quality of the numerical solution. Numerical results are presented to demonstrate the robustness and effectiveness of our $h$ -adaptive algorithm using the double tolerance adaptive strategy.     

A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension

We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials $\{1, x_{i}, x_{i}^{2}\}$ , the shear stress by bilinear polynomials $\{1, x_{i}, x_{j}, x_{i}x_{j}\}$ , and the displacement by linear polynomials $\{1, x_{i} \}$ . The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension $n$ . As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.     

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $\ell ^2$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).     

A New Error Analysis of Crank–Nicolson Galerkin FEMs for a Generalized Nonlinear Schrödinger Equation

In this paper, we study linearized Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. We present the optimal \(L^2\) error estimate without any time-step restrictions, while previous works always require certain conditions on time stepsize. A key to our analysis is an error splitting, in terms of the corresponding time-discrete system, with which the error is split into two parts, the temporal error and the spatial error. Since the spatial error is \(\tau \) -independent, the numerical solution can be bounded in \(L^{\infty }\) -norm by an inverse inequality unconditionally. Then, the optimal \(L^2\) error estimate can be obtained by a routine method. To confirm our theoretical analysis, numerical results in both two and three dimensional spaces are presented.     

The Estimation of Truncation Error by \tau -Estimation for Chebyshev Spectral Collocation Method

In this paper we show how to accurately estimate the local truncation error of the Chebyshev spectral collocation method using $\tau $ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the truncation error. Then, we show the validity of the analysis for the incompressible Navier–Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the truncation error if the precision of the approximations increases with the polynomial order.     

Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

The stochastic collocation method (Babu?ka et al. in SIAM J Numer Anal 45(3):1005–1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411–2442, 2008a; SIAM J Numer Anal 46(5):2309–2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118–1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289–294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229–275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41–44):3187–3206, 2009; Arch Comput Methods Eng 17:435–454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $O(1)$ to moderate dimensions $O(10)$ and to high dimensions $O(100)$ . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.     

A linear system form solution to compute the local space average color

In this document, we present an alternative to the method introduced by Ebner (Pattern Recognit 60–67, 2003; J Parallel Distrib Comput 64(1):79–88, 2004; Color constancy using local color shifts, pp 276–287, 2004; Color Constancy, 2007; Mach Vis Appl 20(5):283–301, 2009) for computing the local space average color. We show that when the problem is framed as a linear system and the resulting series is solved, there is a solution based on LU decomposition that reduces the computing time by at least an order of magnitude.