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This paper is concerned with efficient representations and approximations of the solution to the scattering problem by a system of strongly coupled plasmonic particles. Three schemes are developed: the first is the resonant expansion which uses the resonant modes of the system of particles computed by a conformal transformation, the second is the hybridized resonant expansion which uses linear combinations of the resonant modes for each of the particles in the system as a basis to represent the solution, and the last one is the multipole expansion with respect to the origin. By considering a system formed by two plasmonic particles of circular shape, we demonstrate the relations between these expansion schemes and their advantages and disadvantages both analytically and numerically. In particular, we emphasize the efficiency of the resonant expansion scheme in approximating the near field of the system of particles. The difference between these plasmonic particle systems and the nonresonant dielectric particle system is also highlighted. The paper provides a guidance on the challenges for numerical simulations of strongly coupled plasmonic systems. [ABSTRACT FROM AUTHOR]

Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we first rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we next transform this problem into a coupled family of quasi-periodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasi-periodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretization in the spatial and the quasi periodicity's unit cells. We also provide a simple and efficient way of implementation that does not require numerical integration in the quasi periodicity, together with numerical examples for scattering from locally perturbed periodic surfaces computed by this scheme. [ABSTRACT FROM AUTHOR]

We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation---in the mean parametric norm associated to the elliptic operator---of the error between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the proper orthogonal decomposition (POD) subspaces, except that in our case the norm is parameter-dependent. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the online step, and we prove that the partial sums converge to the continuous solution in the mean parametric elliptic norm. We show that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper. This opens the possibility of computing the PGD expansion by directly solving the optimization problems that yield the optimal subspaces. [ABSTRACT FROM AUTHOR]

In this paper we provide a general framework to study a general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin (SG) approximations, in both incompressible Navier{Stokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19 (2006), pp. 969{998; M. Briant, J. Differential Equations, 259 (2005), pp. 6072{6141] based on hypocoercivity for the deterministic kinetic equations can be easily adopted for sensitivity analysis for random kinetic equations, which gives rise to an exponential convergence of the random solution toward the (deterministic) global equilibrium, under suitable conditions on the collision kernel. Then we use such theory to study the SG methods for the equations, establish hypocoercivity of the SG system and regularity of its solution, and study spectral accuracy and exponential decay in time of the numerical error of the method in a weighted Sobolev norm. [ABSTRACT FROM AUTHOR]

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree k, it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k-1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k-1 are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of H(div)-conformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis, and demonstrate the performance of the method on numerical examples. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates have many applications. As one of the applications we establish best approximations type results with respect to the 퐿푝(0, 푇;퐿2(Ω)) norm for 1 ≤ 푝 ≤ ∞. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty DG methods for spatial discretization. They differ from each other on how the nonlinear term is treated; one of them is based on fully implicit time-stepping and the other uses energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on ϵ-1 only in some low polynomial orders, instead of exponential orders. Similar to [X. Feng and A. Prohl, Numer. Math., 74 (2004), pp. 47-84], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space HM1(Ω) and it is larger than the finite element space. This difficulty is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved by Feng and Prohl. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods. [ABSTRACT FROM AUTHOR]

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical analysis for three-dimensional Bean's critical-state model in type-II superconductivity. We derive hyperbolic mixed variational inequalities of the second kind for the evolution Maxwell equations with Bean's constitutive law between the electric Field and the current density. On the basis of the variational inequality in the magnetic induction formulation, a semidiscrete Ritz-Galerkin approximation problem is rigorously analyzed, and a strong convergence result is proven. Thereafter, we propose a concrete realization of the Ritz-Galerkin approximation through a mixed Finite element method based on edge elements of Nédélec's First family, Raviart-Thomas face elements, divergence-free Raviart-Thomas face elements, and piecewise constant elements. As a Final result, we prove error estimates for the proposed mixed Finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the ith component of the fluid mixture, ci, should be between 0 and 1. However, ci does not satisfy the maximum principle. Therefore, the numerical techniques introduced in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both c1 and c2, respectively, and enforce c1 + c2 = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation. Moreover, ci's are not the conservative variables; as a result, the classical bound-preserving limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091-3120] cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in the L∞-norm and good performance of the numerical technique. [ABSTRACT FROM AUTHOR]

Over the past few decades there has been a strong effort toward the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this postprocessor. These advantages can be exploited during flow visualization, for example, by applying the SIAC filter to DG data before streamline computations [M. Steffen, S. Curtis, R. M. Kirby, and J. K. Ryan, IEEE Trans. Vis. Comput. Graphics, 14 (2008), pp. 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [D. Walfisch, J. K. Ryan, R. M. Kirby, and R. Haimes, J. Sci. Comput., 38 (2009), pp. 164-184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure, and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community. [ABSTRACT FROM AUTHOR]

In this paper, we develop a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semiconductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity (uniform in the Knudsen number) of the solution in the random space and prove the spectral accuracy of the gPC-SG method. We then use the asymptotic-preserving framework for the deterministic counterpart developed in [S. Jin and L. Pareschi, J. Comput. Phys., 161 (2000), pp. 312-330] to come up with the stochastic asymptotic-preserving gPC-SG method for the problem under study which is efficient in the diffusive regime. Numerical experiments are conducted to validate the accuracy and asymptotic properties of the method. [ABSTRACT FROM AUTHOR]

We analyze a second order pure Lagrange-Galerkin method for variable coefficient convection-diffusion (possibly degenerate-diffusion) equations with mixed Dirichlet--Robin boundary conditions. In a previous paper the proposed second order pure Lagrangian time discretization scheme was introduced and analyzed for the same problem. More precisely, the l∞(H¹) stability and l∞(H¹) error estimates of order O(Δt²) had been obtained. Moreover, for the particular case of incompressible flows, stability inequalities with constants independent of the final time have been stated. In the present paper ∞(H¹) error estimates of order O(Δt²) + O(hk) are obtained for the fully discretized pure Lagrange-Galerkin method. To prove these results we use some properties obtained in the previous paper. Finally, numerical tests are presented that confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The numerical solution of time-dependent radiative transfer problems is challenging, due to the high dimension and to the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and also to preserve basic properties such as exponential stability and decay to equilibrium on the discrete level. We present the basic a priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure to first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed PN-finite element time stepping scheme is demonstrated by computational results. [ABSTRACT FROM AUTHOR]

The positivity-preserving property is an important and challenging issue for the numerical solution of radiative transfer equations. In the past few decades, different numerical techniques have been proposed to guarantee positivity of the radiative intensity in several schemes; however it is difficult to maintain both high order accuracy and positivity. The discontinuous Galerkin (DG) finite element method is a high order numerical method which is widely used to solve the neutron/photon transfer equations, due to its distinguished advantages such as high order accuracy, geometric exibility, suitability for h-and p-adaptivity, parallel efficiency, and a good theoretical foundation for stability and error estimates. In this paper, we construct arbitrarily high order accurate DG schemes which preserve positivity of the radiative intensity in the simulation of both steady and unsteady radiative transfer equations in one- and two-dimensional geometry by using a combined technique of the scaling positivity-preserving limiter in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 8918-8934] and a new rotational positivity-preserving limiter. This combined limiter is simple to implement and we prove the properties of positivity-preserving and high order accuracy rigorously. One- and two-dimensional numerical results are provided to verify the good properties of the positivity-preserving DG schemes. [ABSTRACT FROM AUTHOR]

This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin (SIPG) method for the numerical solution of optimal control problems governed by linear reaction-advection-diffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advection-dominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advection-dominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too "weak" to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions, which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results. [ABSTRACT FROM AUTHOR]

A full discrete two-level postprocessing Fourier Galerkin scheme for the unsteady Navier-Stokes equations with a periodic boundary condition is proposed in this paper. By defining a new projection, the interaction between the large and small eddies is reflected by the associated space splitting to some extent. Therefore, a weakly coupled system of the large and small eddies is obtained. Stability and error estimates for the weakly coupled two-level scheme are established in the paper. The proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse-level subspace by solving the coarse-level standard Galerkin equation, and the matrix of the linear algebraic equations arising in each fine-level time stepping is a sparse matrix, especially for large scale computations. As a result, the new scheme saves a lot of CPU time and memory in comparison with the standard Fourier Galerkin method for deriving an approximation of prescribed accuracy. [ABSTRACT FROM AUTHOR]

In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions. [ABSTRACT FROM AUTHOR]

In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for space-time discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable. [ABSTRACT FROM AUTHOR]

This paper is the second part of our work on a priori error analysis for finite element discretizations of parabolic optimal control problems. In the first part [SIAM J. Control Optim., 47 (2008), pp. 1150-1177] problems without control constraints were considered. In this paper we derive a priori error estimates for space-time finite element discretizations of parabolic optimal control problems with pointwise inequality constraints on the control variable. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For the treatment of the control discretization we discuss different approaches, extending techniques known from the elliptic case. [ABSTRACT FROM AUTHOR]

This paper is the third and last part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The purpose of this paper is to extend the framework that has been developed in Part II for two-field Friedrichs' systems associated with second-order PDEs. We now consider two-field Friedrichs' systems with partial L2-coercivity and three-field Friedrichs' systems with an even weaker L2-coercivity hypothesis. In particular, this work generalizes the discontinuous Galerkin methods of Part II to compressible and incompressible linear continuum mechanics. We also show how the stabilizing parameters of the method must be set when the two-field Friedrichs' system is composed of terms that may be of different magnitude, thus accounting, for instance, for advection-diffusion equations at high Péclet numbers. [ABSTRACT FROM AUTHOR]

This is the third part of a series of papers on least-squares Galerkin methods for parabolic initial-boundary value problems. These methods are based on the minimization of a least-squares functional for an equivalent first-order system over space and time with respect to suitable discrete spaces. This paper presents the derivation and analysis of one-step methods for semidiscretization in time from least-squares principles for semilinear parabolic problems. One of the most important features of the least-squares methodology is a built-in a posteriori estimate for the approximation error. For the presentation in this paper, we focus our attention on the specific combination of piecewise linear, not necessarily continuous, functions in time with continuous piecewise linear for the flux and scalar variables, respectively. For the resulting method, a convergence result is shown for the scalar variable. [ABSTRACT FROM AUTHOR]

Summation-by-parts (SBP) finite-difference discretizations share many attractive properties with Galerkin finite-element methods (FEMs), including time stability and superconvergent functionals; however, unlike FEMs, SBP operators are not completely determined by a basis, so the potential exists to tailor SBP operators to meet different objectives. To date, application of highorder SBP discretizations to multiple dimensions has been limited to tensor product domains. This paper presents a definition for multidimensional SBP finite-difference operators that is a natural extension of one-dimensional SBP operators. Theoretical implications of the definition are investigated for the special case of a diagonal-norm (mass) matrix. In particular, a diagonal-norm SBP operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial-basis matrix has full rank when evaluated at the cubature nodes. Appropriate simultaneous-approximation terms are developed to impose boundary conditions weakly, and the resulting discretizations are shown to be time stable. Concrete examples of multidimensional SBP operators are constructed for the triangle and tetrahedron; similarities and differences between these SBP operators and those typically used in spectral-element, spectral-difference, and nodal discontinuous Galerkin methods are discussed. An assembly process is described that builds SBP operators on a global domain from element-level operators. Numerical results of linear advection on a doubly periodic domain demonstrate the accuracy and time stability of the simplex operators. [ABSTRACT FROM AUTHOR]

Convolving the output of discontinuous Galerkin computations with symmetric smoothness-increasing accuracy-conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. This paper interprets these filters as instances of a general class of position-dependent (PSIAC) spline filters that can have nonuniform knot sequences and skip B-splines of the sequence. PSIAC filters with rational knot sequences have rational coefficients. For prototype knot sequences, such as integer sequences that may have repeated entries, PSIAC filters can be expressed in symbolic form. Based on the insight that filters for shifted or scaled knot sequences are easily derived by nonuniform scaling of one prototype filter, a single filter can be reused in different locations and at different scales. Computing a value of the convolution then simplifies to forming a scalar product of a short vector with the local output data. Restating one-sided filters in this form improves both stability and efficiency compared to their original formulation via numerical integration. PSIAC filtering is demonstrated for several established and one new boundary filter. [ABSTRACT FROM AUTHOR]

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the L∞ norm. The discretization method uses continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows one to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [Numer. Math., submitted; available online at http://arxiv.org/abs/1505.04808/]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish an interior best approximation property that shows a more local behavior of the error at a given point. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an optimal preconditioner for a general linear symmetric positive definite (SPD) system by following the basic idea of the well-known BPX framework. The SPD system arises from a large number of nonstandard finite element methods for diffusion problems, including the well-known hybridized Raviart-Thomas and Brezzi-Douglas-Marini mixed element methods, the hybridized discontinuous Galerkin method, the weak Galerkin method, and the nonconforming Crouzeix-Raviart element method. We prove that the presented preconditioner is optimal, in the sense that the condition number of the preconditioned system is independent of the mesh size. Numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a symplectic model reduction technique, proper symplectic decomposition (PSD) with symplectic Galerkin projection, is proposed to save the computational cost for the simplification of large-scale Hamiltonian systems while preserving the symplectic structure. As an analogy to the classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is designed to build a symplectic subspace to fit empirical data, while the symplectic Galerkin projection constructs a reduced Hamiltonian system on the symplectic subspace. For practical use, we introduce three algorithms for PSD, which are based upon the cotangent lift, complex singular value decomposition, and nonlinear programming. The proposed technique has been proven to preserve system energy and stability. Moreover, PSD can be combined with the discrete empirical interpolation method to reduce the computational cost for nonlinear Hamiltonian systems. Owing to these properties, the proposed technique is better suited than the classical POD-Galerkin approach for model reduction of Hamiltonian systems, especially when long-time integration is required. The stability, accuracy, and efficiency of the proposed technique are illustrated through numerical simulations of linear and nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we present a local discontinuous Galerkin (LDG) method and two unconditionally energy stable schemes for the phase field crystal (PFC) equation. The semidiscrete energy stability of the LDG method is proved first. The PFC equation is a sixth order nonlinear partial differential equation (PDE), which leads to the severe time step restriction (Δt = O(Δx6)) of explicit time discretization methods to maintain stability. Due to this, we introduce semi-implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities. To improve the temporal accuracy, the spectral deferred correction (SDC) method and a high order semi-implicit Runge-Kutta method combining with the first-order convex splitting method are adopted for the PFC equation with constant and degenerate mobility, respectively. The equations at the implicit time level are nonlinear and we employ an efficient nonlinear multigrid solver to solve the equations. In particular, we show the multigrid solver has optimal complexity numerically. Numerical results are also given to illustrate that the combination of the LDG method for spatial approximation, SDC, and high order semi-implicit time marching methods with the multigrid solver provides an efficient and practical approach when solving the PFC equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose to apply the parametrized maximum-principle-preserving (MPP) flux limiter in [T. Xiong, J.-M. Qiu, and Z. Xu, J. Comput. Phys., 252 (2013), pp. 310-331] to the discontinuous Galerkin (DG) method for solving the convection-diffusion equations. The feasibility of applying the proposed MPP flux limiters is based on the fact that cell averages of the DG solutions are updated in a conservative fashion (by using flux difference). Compared with the earlier approach in preserving the MPP property of DG solutions for convection-diffusion problems [Y. Zhang, X. Zhang, and C.-W. Shu, J. Comput. Phys., 234 (2012), pp. 295-316], we avoid the difficulty of rewriting cell averages as a convex combination of point values in the presence of diffusion terms for DG schemes of higher than second order accuracy. As a result, our proposed parametrized MPP flux limiter can be applied to DG methods of arbitrary high order. We theoretically prove that up to the third order accuracy can be preserved for scalar one-dimensional problems. Numerical evidence is presented to show that the proposed MPP flux limiter method does not adversely affect the desired high order accuracy, nor does it require restrictive time steps. Numerical experiments including those for incompressible flow examples demonstrate the high order accuracy preserving, the MPP performance, and the robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we consider a proper generalized decomposition method to solve the steady incompressible Navier--Stokes equations with random Reynolds number and forcing term. The aim of such a technique is to compute a low-cost reduced basis approximation of the full stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of preexisting deterministic Navier--Stokes solvers. It has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m-dimensional reduced basis rather than M coupled problems of the full stochastic Galerkin approximation space, with m « M (up to one order of magnitude for the problem at hand in this work). [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin (DG) method has very quickly found utility in such diverse applications as computational solid mechanics, fluid mechanics, acoustics, and electromagnetics. The DG methodology merely requires weak constraints on the fluxes between elements. This feature provides a flexibility which is difficult to match with conventional continuous Galerkin methods. However, allowing discontinuity between element interfaces can in turn be problematic during simulation postprocessing, such as in visualization. Consequently, smoothness-increasing accuracyconserving (SIAC) filters were proposed in [M. Steffen et al., IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 680-692, D. Walfisch et al., J. Sci. Comput., 38 (2009), pp. 164-184] as a means of introducing continuity at element interfaces while maintaining the order of accuracy of the original input DG solution. Although the DG methodology can be applied to arbitrary triangulations, the typical application of SIAC filters has been to DG solutions obtained over translation invariant meshes such as structured quadrilaterals and triangles. As the assumption of any sort of regularity including the translation invariance of the mesh is a hindrance towards making the SIAC filter applicable to real life simulations, we demonstrate in this paper for the first time the behavior and complexity of the computational extension of this filtering technique to fully unstructured tessellations. We consider different types of unstructured triangulations and show that it is indeed possible to get reduced errors and improved smoothness through a proper choice of kernel scaling. These results are promising, as they pave the way towards a more generalized SIAC filtering technique. [ABSTRACT FROM AUTHOR]

This paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A finite volume scheme is used for the deterministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with a Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic mul-tiresolution analysis. Efficiency of the overall adaptive scheme is assessed on a stochastic nonlinear conservation law with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in the context of problems involving a large number of stochastic parameters. [ABSTRACT FROM AUTHOR]

We propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper. The central objective is to develop for such classes adaptive solution concepts with provable error reduction. To adaptively resolve anisotropic solution features such as propagating singularities, the presently proposed variational formulations allow, in particular, the employment of trial spaces spanned by directional representation systems. Since such systems, typically given as frames, are known to be stable only in L2, special emphasis is placed on L2-stable formulations. The proposed stability concept is based on perturbations of certain "ideal" test spaces in Petrov--Galerkin formulations; see also [L. F. Demkowicz and J. Gopalakrishnan, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1558-1572], [L. Demkowicz and J. Gopalakrishnan, Numer. Methods Partial Differential Equations, 27 (2011), pp. 70-105], [J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V. Calo, J. Comput. Phys., 230 (2011), pp. 2406-2432]. We propose a general strategy for realizing the resulting Petrov-Galerkin schemes based on an Uzawa iteration circumventing an excessively expensive computation of corresponding test basis functions. Moreover, based on this iteration, we develop adaptive solution concepts with provable error reduction. The results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we construct an "interior penalty" discontinuous Galerkin method to approximate the minimizer of a variational problem related to the p(x)-Laplacian. The function p : Ω → [p1,p2] is log-Hölder continuous and 1 < p1 ≤ p2 < ∞. We prove that the minimizers of the discrete functional converge to the solution. We also make some numerical experiments in dimension one to compare this method with the conforming Galerkin method, in the case where p1 is close to one. This example is motivated by its applications to image processing. [ABSTRACT FROM AUTHOR]

This paper is concerned with the superconvergence of the discoutinuous Galerkin solutions for delay differential equations with proportional delays vanishing at t = 0. Two types of superconvergenee are analyzed here. The first is based oil interpolation postprocessing to improve the global convergence order by finding the superconvergence points of discontinuous Galerkin solutions. The second type follows from the integral iteration which just requires a local integration procedure applied to the discontinuous Galerkin solution, thus increasing the order of convergence. The theoretical results are illustrated by a broad range of numerical examples. [ABSTRACT FROM AUTHOR]

In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 8191-8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L¹ norm and a very fast convergence speed, but only first order accuracy in the L∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples. [ABSTRACT FROM AUTHOR]

Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering is a promising technique not only ill improving the order of the numerical solution obtained by a discontinuous Galerkin (DG) method but also in increasing the smoothness of the field and improving the magnitude of the errors. This was initially established as an accuracy enhancement technique by Cockburn et al. for linear hyperbolic equations to handle smooth solutions [Math. Comp., 72 (2003), pp. 577 606]. By implementing this technique, the quality of the solution can be improved from order k + 1 to order 2k + 1 in the L2-norm. Ryan and Shu used these ideas to extend this technique to be able to handle postprocessing near boundaries ms well as discontinuities [Methods Appl. Anal., 10 (2003), pp. 295-307]. However, this presented difficulties as the resulting error had a stair-stepping effect and the errors themselves were not improved over those of the DG solution unless the mesh was suitably refined. In this paper, we discuss an improved filter for enhancing DG solutions that easily switches between one-sided postprocessing to handle boundaries or discontinuities and symmetric postprocessing for smooth regions. We numerically demonstrate that the magnitude of the errors using the modified postprocessor is roughly the same ms that of the errors for the symmetric postprocessor itself, regardless of the boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability arid convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed. [ABSTRACT FROM AUTHOR]

Variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena, most notably in weather prediction, and are usually solved via an ill-posed optimal control problem for the initial state at the time of the first available measurements. An alternative "forward" approach focuses on computation of the final state after this interval-which is just as suitable for prediction purposes-and is well-posed without additional regularization. Specifically, it is possible to compute projections of the unknown final state on all elements of an orthonormal basis, which theoretically allows for the complete reconstruction of the final state. In this paper, an efficient numerical method for linear evolution equations of diffusive type is presented, and convergence of the numerical approximation based on a discontinuous Galerkin discretization is proved. The key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition. Numerical examples for a scalar convection-diffusion equation in two and three dimensions show the effectiveness of the method. [ABSTRACT FROM AUTHOR]

This paper presents a locally conservative finite element method based on enriching the approximation space of the continuous Galerkin method with elementwise constant functions. The proposed method has a smaller number of degrees of freedom than the discontinuous Galerkin method. Numerical examples on coupled flow and transport in porous media are provided to illustrate the advantages of this method. We also present a theoretical analysis of the method and establish optimal convergence of numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation and its zero-viscosity and zero-dispersion limits. The energy stability for general solutions are proved, and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to illustrate the accuracy and capability of the LDG method. The zero-viscosity and zero-dispersion properties of the HS equation are studied in a numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze two augmented mixed finite element methods for a velocity-pressure-stress formulation of the stationary Stokes equations. Our approach, which extends analog results for linear elasticity problems, is based on the introduction of the Galerkin least-squares-type terms arising from the constitutive and equilibrium equations and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms, whence the associated Galerkin schemes become well-posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and Raviart-Thomas elements for the stresses, thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation. Alternatively, the above factor reduces to 4 when a second augmented variational formulation, involving only the velocity and the stress as unknowns, is employed. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element schemes. Finally, extensive numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. [ABSTRACT FROM AUTHOR]

The numerical solution of large-scale continuous-time Lyapunov matrix equations is of great importance in many application areas. Assuming that the coefficient matrix is positive definite, but not necessarily symmetric, in this paper we analyze the convergence of projection-type methods for approximating the solution matrix. Under suitable hypotheses on the coefficient matrix, we provide new asymptotic estimates for the error matrix when a Galerkin method is used in a Krylov subspace. Numerical experiments confirm the good behavior of our upper bounds when linear convergence of the solver is observed. [ABSTRACT FROM AUTHOR]

This paper deals with superconvergence phenomena in general grids when projectionbased approximations are used for solving Fredholm integral equations of the second kind with weakly singular kernels. Four variants of the Galerkin method are considered. They are the classical Galerkin method, the iterated Galerkin method, the Kantorovich method, and the iterated Kantorovich method. It is proved that the iterated Kantorovich approximation exhibits the best superconvergence rate if the right-hand side of the integral equation is nonsmooth. All error estimates are derived for an arbitrary grid without any uniformity or quasi-uniformity condition on it, and are formulated in terms of the data without any additional assumption on the solution. Numerical examples concern the equation governing transfer of photons in stellar atmospheres. The numerical results illustrate the fact that the error estimates proposed in the different theorems are quite sharp, and confirm the superiority of the iterated Kantorovich scheme. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

The stress-displacement-pressure formulation of the elasticity problem may suffer from two types of numerical instabilities related to the finite element interpolation of the unknowns. The first is the classical pressure instability that occurs when the solid is incompressible, whereas the second is the lack of stability in the stresses. To overcome these instabilities, there are two options. The first is to use different interpolation for all the unknowns satisfying two inf-sup conditions. Whereas there are several displacement-pressure interpolations that render the pressure stable, less possibilities are known for the stress interpolation. The second option is to use a stabilized finite element formulation instead of the plain Galerkin approach. If this formulation is properly designed, it is possible to use arbitrary interpolation for all the unknowns. The purpose of this paper is precisely to present one of such formulations. In particular, it is based on the decomposition of the unknowns into their finite element component and a subscale, which will be approximated and whose goal is to yield a stable formulation. A singular feature of the method to be presented is that the subscales will be considered orthogonal to the finite element space. We describe the design of the formulation and present the results of its numerical analysis. [ABSTRACT FROM AUTHOR]

We develop a family of new interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. This model is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It has been recently shown that the convective part of this system is of a mixed hyperbolic-elliptic-type, which may cause severe instabilities when the studied system is solved by straightforward numerical methods. Therefore, the first step in the derivation of our new methods is made by introducing the new variable for the gradient of the chemoattractant concentration and by reformulating the original Keller-Segel model in the form of a convection-diffusionreaction system with a hyperbolic convective part. We then design interior penalty discontinuous Galerkin methods for the rewritten Keller-Segel system. Our methods employ the central-upwind numerical fluxes, originally developed in the context of finite-volume methods for hyperbolic systems of conservation laws. In this paper, we consider Cartesian grids and prove error estimates for the proposed high-order discontinuous Galerkin methods. Our proof is valid for pre-blow-up times since we assume boundedness of the exact solution. We also show that the blow-up time of the exact solution is bounded from above by the blow-up time of our numerical solution. In the numerical tests presented below, we demonstrate that the obtained numerical solutions have no negative values and are oscillation-free, even though no slope-limiting technique has been implemented. [ABSTRACT FROM AUTHOR]

In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signaling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equation, known as the chemical master equation (CME). Until now, the CME has primarily been treated by Monte Carlo techniques, the most prominent of which is the stochastic simulation algorithm [D. T. Gillespie, J. Comput. Phys., 22 (1976), pp. 403-434]. The paper presents an alternative, which focuses on the discrete partial differential equation (PDE) structure of the CME. This allows us to adopt ideas from adaptive discrete Galerkin methods as first suggested by Deuflhard and Wulkow [IMPACT Comput. Sci. Engrg., 1 (1989), pp. 269-301] for polyreaction kinetics and independently developed by Engblom. From the two different options for discretizing the CME as a discrete PDE, Engblom chose the method of lines approach (first space, then time), whereas we strongly advocate use of the Rothe method (first time, then space) for clear theoretical and algorithmic reasons. Numerical findings at two rather challenging problems illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further improvement of the method worked out here. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Camassa-Holm equation which contains nonlinear high-order derivatives. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L2 stability for general solutions and give a detailed error estimate for smooth solutions, and provide numerical simulation results for different types of solutions of the nonlinear Camassa-Holm equation to illustrate the accuracy and capability of the LDG method. [ABSTRACT FROM AUTHOR]

We propose in this paper a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis. The compression leads to a sparse matrix with only O(n log n) nonzero entries, where 2n or 2n + 1 denotes the order of the matrix. Based on this compression strategy, we develop a fast Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the fast Fourier-Galerkin method gives the optimal convergence order O(n-t), where t denotes the degree of regularity of the exact solution. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We show that solving this system requires only an O(n log2 n) number of multiplications. We present numerical examples to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]