Piecewise Polynomials Of Degree Research Articles

Superconvergence of Arbitrary Lagrangian--Eulerian Discontinuous Galerkin Methods for Linear Hyperbolic Equations

In this paper, we study the superconvergence properties of an arbitrary Lagrangian--Eulerian discontinuous Galerkin (ALE-DG) method with approximations to one-dimensional linear hyperbolic equations. The ALE-DG method is a mesh moving method; we need to deal with the new difficulties brought by the time-dependent test function space and grid velocity field. Since the time derivative cannot commute with the space projections for the ALE-DG method, we will introduce the material derivative in our analysis. With the help of the scaling argument and material derivative, we build a special interpolation function by constructing the correction functions and prove that the numerical solution is superclose to the interpolation function in the $L^2$-norm. The order of the superconvergence is 2k+1 when piecewise polynomials of degree at most $k$ are used. We also rigorously prove a (2k+1)th-order superconvergence rate for the domain and cell average and at the downwind points in the maximal and average norm. Furthermore, we prove that the function value approximation is superconvergent with a rate $k+2$ at the right Radau points and a superconvergence rate $k+1$ for the derivative approximation at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.

A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first order optimality conditions, and regularity estimates for the optimal variables. We propose two strategies to discretize the fractional optimal control problem: a semidiscrete approach where the control is not discretized - the so-called variational discretization approach - and a fully discrete approach where the control variable is discretized with piecewise constant functions. Both schemes rely on the discretization of the state equation with the finite element space of continuous piecewise polynomials of degree one. We derive a priori error estimates for both solution techniques. We illustrate the theory with two-dimensional numerical tests.

Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems

In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

Superconvergence of discontinuous Galerkin method with interior penalties for singularly perturbed two-point boundary-value problems

In this paper, superconvergence properties of the discontinuous Galerkin method for singularly perturbed two-point boundary-value problems of reaction–diffusion and convection–diffusion types are studied. By using piecewise polynomials of degree k on modified Shishkin mesh, superconvergence error bounds of $$(N^{-1}\ln N)^{k+1}$$ in the discrete energy norm are established, where N is the number of elements. Finally, the convergence result is verified numerically.

Guided wave propagation in a multilayered magneto-electro-elastic curved panel by Chebyshev spectral elements method

This paper investigates the dispersion characteristics of guided waves in a multilayered magneto-electro-elastic curved panel. Based on the Hamilton principle, the Chebyshev spectral element method is applied along the radial and circumferential directions to obtain the dispersion equation of the waves. The advantage of the spectral element method over a high degree piecewise polynomial basis function is the high order of accuracy. The influences on the dispersion characteristics is analyzed in terms of four factors: structural radian, magneto-electro effect, thickness-diameter ratio, and stacking sequence. In addition, the dispersion curves in the curved panel display a phenomenon which different from plate and hollow cylinders: mode conversion. We explain this from the perspective of physics and mathematics, separately. The structural radian and thickness-diameter ratio show the most significant effect on the mode conversion.

Finite element approximation of reaction–diffusion problems using an exponentially graded mesh

We present the analysis of an h version Finite Element Method for the approximation of the solution to singularly perturbed reaction–diffusion problems posed in smooth domains Ω⊂R2. The method uses piecewise polynomials of degree p in each variable, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We establish robust, optimal convergence rates in a variety of norms and illustrate our theoretical findings through numerical computations.

A Hybridizable Discontinuous Galerkin Method for Kirchhoff Plates

With the introduction of numerical traces respectively related to the normal bending moment, the twisting moment and the effective transverse shear force, and based on the Hermann–Miyoshi formulation, this paper proposes a hybridizable discontinuous Galerkin (HDG) method for Kirchhoff plate bending problems. The piecewise polynomials of degrees $$k-1$$ and k are used to approximate the moment and the deflection, respectively. The optimal and superconvergent error estimates are derived under minimal regularity assumptions on the exact solution. The key ingredients in the analysis include the derivation of a discrete inf-sup condition and some local lower bound estimates of a posteriori error analysis. The significant feature of the HDG method is superconvergence as well as the low number of globally coupled degrees of freedom associated with Lagrange multipliers. Furthermore, a new discrete deflection is constructed by postprocessing the solution of the HDG method, which superconverges to the deflection with order $$k+1$$ in broken $$H^1$$ norm. Finally, some numerical results are shown to demonstrate the theoretical results.

An improved algorithm based on deviation of the error estimation for first-order integro-differential equations

In this paper, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of first-order Fredholm–Volterra integro-differential equations. In the first step, we find the deviation of the error for Fredholm–Volterra integro-differential equations by using defect correction principle. Then we show that for m degree piecewise polynomial collocation method, our method provides order $$\mathcal {O}(h^{m+1})$$ for the deviation of the error. Also we improve the piecewise polynomial collocation method by using the deviation of the error estimation. Numerical results in the last section are included to confirm the theoretical results.

Deviation of the error estimation for Volterra integro-differential equations

In this paper, we study an efficient asymptotically correction of a-posteriori error estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integro-differential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O(hm+1) as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.

A posteriori error estimation for an augmented mixed-primal method applied to sedimentation–consolidation systems

In this paper we develop the a posteriori error analysis of an augmented mixed-primal finite element method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation–consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection – nonlinear diffusion equation describing the transport of the solids volume fraction. We derive two efficient and reliable residual-based a posteriori error estimators for a finite element scheme using Raviart–Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤k+1 for both velocity and concentration. For the first estimator we make use of suitable ellipticity and inf–sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clément interpolant and Raviart–Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle-bubble and edge-bubble functions. Next, we analyse an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decompositions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method.

Conservative High Order Positivity-Preserving Discontinuous Galerkin Methods for Linear Hyperbolic and Radiative Transfer Equations

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems

In this paper, we present and analyze a superconvergent local discontinuous Galerkin (LDG) scheme for the numerical solution of nonlinear KdV-type partial differential equations. Optimal a priori error estimates for the LDG solution and for the two auxiliary variables that approximate the first-and second-order derivative are derived in the L2-norm for the semi-discrete formulation. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order p+1 towards the derivative of a special projection of the exact solution. We use this results to prove that the LDG solution is superconvergent with order p+3/2 toward a special Gauss-Radau projection of the exact solution. Finally, several numerical examples are given to validate the theoretical results. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p≥1 and under the condition that |f'(u)| possesses a uniform positive lower bound, where f(u) is the nonlinear flux function. Our experiments demonstrate that our results hold true for KdV equations with general flux functions.

A locking free Reissner-Mindlin element with weak Galerkin rotations

A locking free finite element method is developed for the Reissner-Mindlin equations in their primary form. In this method, the transverse displacement is approximated by continuous piecewise polynomials of degree $k+1$ and the rotation is approximated by weak Galerkin elements of degree $k$ for $k≥1$ . A uniform convergence in thickness of the plate is established for this finite element approximation. The numerical examples demonstrate locking free of the method.

Asymptotically Exact Posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Nonlinear Convection–Diffusion Problems

In this paper, we present and analyze implicit a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to nonlinear convection–diffusion problems in one space dimension. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the $$L^2$$ -norm for the semi-discrete formulation. More precisely, we identify special numerical fluxes and a suitable projection of the initial condition for the LDG scheme to achieve $$p+1$$ order of convergence for the solution and its spatial derivative in the $$L^2$$ -norm, when piecewise polynomials of degree at most p are used. We further prove that the derivative of the LDG solution is superconvergent with order $$p+1$$ towards the derivative of a special projection of the exact solution. We use this result to prove that the LDG solution is superconvergent with order $$p+3/2$$ towards a special Gauss–Radau projection of the exact solution. Our superconvergence results allow us to show that the leading error term on each element is proportional to the $$(p+1)$$ -degree right Radau polynomial. We use these results to construct asymptotically exact a posteriori error estimator. Furthermore, we prove that the a posteriori LDG error estimate converges at a fixed time to the true spatial error in the $$L^2$$ -norm at $$\mathcal {O}(h^{p+3/2})$$ rate. Finally, we prove that the global effectivity index in the $$L^2$$ -norm converge to unity at $$\mathcal {O}(h^{1/2})$$ rate. Our proofs are valid for arbitrary regular meshes using $$P^p$$ polynomials with $$p\ge 1$$ . Finally, several numerical examples are given to validate the theoretical results.

Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection

This paper presents error analysis of hybridizable discontinuous Galerkin (HDG) time-domain method for solving time dependent Schrödinger equations. The numerical trace and numerical flux are constructed to preserve the conservative property for the density of the particle described. We prove that there exist the superconvergence properties of the HDG method, which do hold for second-order elliptic problems, uniformly in time for the semidiscretization by the same method of Schrödinger equations provided that enough regularity is satisfied. Thus, if the approximations are piecewise polynomials of degreer, the approximations to the wave function and the flux converge with orderr+ 1. The suitably chosen projection of the wave function into a space of lower polynomial degree superconverges with orderr+ 2 forr≥ 1 uniformly in time. The application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential convergence with orderr+ 2 forr≥ 1 inL2is also uniformly in time.

A stochastic local discontinuous Galerkin method for stochastic two-point boundary-value problems driven by additive noises

The local discontinuous Galerkin (LDG) method has been successfully applied to deterministic boundary-value problems (BVPs) arising from a wide range of applications. In this paper, we propose a stochastic analogue of the LDG method for stochastic two-point BVPs. We first approximate the white noise process by a piecewise constant random process to obtain an approximate BVP. We show that the solution of the new BVP converges to the solution of the original problem. The new problem is then discretized using the LDG method for deterministic problems. We prove that the solution to the new approximate BVP has better regularity which facilitates the convergence proof for the proposed LDG method. More precisely, we prove L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be two in the mean-square sense, when piecewise polynomials of degree at most p are used. Finally, several numerical examples are provided to illustrate the theoretical results.

A priori and a posteriori error analysis of an augmented mixed-FEM for the Navier–Stokes–Brinkman problem

We introduce and analyze an augmented mixed finite element method for the Navier–Stokes–Brinkman problem with nonsolenoidal velocity. We employ a technique previously applied to the stationary Navier–Stokes equation, which consists of the introduction of a modified pseudostress tensor relating the gradient of the velocity and the pressure with the convective term, and propose an augmented pseudostress–velocity formulation for the model problem. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Banach fixed point theorem, combined with the Lax–Milgram lemma, are applied to prove the unique solvability of the continuous and discrete systems. We point out that no discrete inf–sup conditions are required for the solvability analysis, and hence, in particular for the Galerkin scheme, arbitrary finite element subspaces of the respective continuous spaces can be utilized. For instance, given an integer k≥0, the Raviart–Thomas spaces of order k and continuous piecewise polynomials of degree ≤k+1 constitute feasible choices of discrete spaces for the pseudostress and the velocity, respectively, yielding optimal convergence. We also emphasize that, since the Dirichlet boundary condition becomes a natural condition, the analysis for both the continuous an discrete problems can be derived without introducing any lifting of the velocity boundary datum. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed method. The proof of reliability makes use of a global inf–sup condition, a Helmholtz decomposition, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, approximation properties of the L2-orthogonal projector, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results illustrating the performance of the augmented mixed method, confirming the theoretical rate of convergence and properties of the estimator, and showing the behavior of the associated adaptive algorithms, are reported.

Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations

In this paper, we study the superconvergence behavior of the semi-discrete discontinuous Galerkin (DG) method for scalar nonlinear hyperbolic equations in one spatial dimension. Superconvergence results for problems with fixed and alternating wind directions are established. On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of $2k+1$ when upwind fluxes and piecewise polynomials of degree $k$ are used. Moreover, we also prove that the function value approximation of the DG solution is superconvergent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order of k+2 and k+1, respectively. As a byproduct, we show a (k+2)th order superconvergence of the DG solution towards the Gauss--Radau projection of the exact solution. On the other hand, superconvergence resu...

Numerical analysis of a dual-mixed problem in non-standard Banach spaces

In this paper we analyze the numerical approximation of a saddle-point problem posed innon-standard Banach spaces $\mathrm{H}(\mathrm{div}_{p}\,, \Omega)\times L^q(\Omega)$, where$\mathrm{H}(\mathrm{div}_{p}\,, \Omega):= \{{\boldsymbol\tau} \in [L^2(\Omega)]^n \colon \mathrm{div} {\boldsymbol\tau} \in L^p(\Omega)\},$with $p>1$ and $q\in \mathbb{R}$ being the conjugate exponent of $p$ and $\Omega\subseteq \mathbb{R}^n$ ($n\in\{2,3\}$)a bounded domain with Lipschitz boundary $\Gamma$. In particular, we are interestedin deriving the stability properties of the forms involved (inf-sup conditions, boundedness),which are the main ingredients to analyze mixed formulations. In fact, by using these propertieswe prove the well-posedness of the corresponding continuous and discrete saddle-point problems by means of theclassical Babuska-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomaselements of order $k\geq 0$ combined with piecewise polynomials of degree $k$. In addition weprove optimal convergence of the numerical approximation in the associated Lebesgue norms.Next, by employing the theory developed for the saddle-point problem, weanalyze a mixed finite element method for a convection-diffusion problem, providing well-posedness of the continuousand discrete problems and optimal convergence under a smallness assumption on the convective vector field.Finally, we corroborate the theoretical results with suitable numerical results in two and three dimensions.

Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations

We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.