Tensor Product Polynomials Of Degree Research Articles

An adjoint-based super-convergent Galerkin approximation of eigenvalues

We present a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations. We consider the eigenvalue as a non-linear functional of its corresponding eigenfunction and show how to extend the adjoint-based approach proposed in Cockburn and Wang (2017) [14], to compute it. We illustrate the method on a second-order elliptic eigenvalue problem. Our extensive numerical results show that the approximate eigenvalues computed by our method converge with a rate of 4k+2 when tensor-product polynomials of degree k are used for the Galerkin approximations. In contrast, eigenvalues obtained by standard finite element methods such as the mixed method or the discontinuous Galerkin method converge with a rate at most of 2k+2. We present numerical results which show the performance of the method for the classic unit square and L-shaped domains, and for the quantum harmonic oscillator. We also present experiments uncovering a new adjoint-corrected approximation of the eigenvalues provided by the hybridizable discontinuous Galerkin method which converges with order 2k+2, as well as results showing the possibilities and limitations of using the adjoint-correction term as an asymptotically exact a posteriori error estimate.

Two Efficient and Reliable a posteriori Error Estimates for the Local Discontinuous Galerkin Method Applied to Linear Elliptic Problems on Cartesian Grids

In this paper, we derive two a posteriori error estimates for the local discontinuous Galerkin (LDG) method applied to linear second-order elliptic problems on Cartesian grids. We first prove that the gradient of the LDG solution is superconvergent with order $$p+1$$ towards the gradient of Gauss-Radau projection of the exact solution, when tensor product polynomials of degree at most p are used. Then, we prove that the gradient of the actual error can be split into two parts. The components of the significant part can be given in terms of $$(p+1)$$ -degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We further develop a postprocessing gradient recovery scheme for the LDG solution. This recovered gradient superconverges to the gradient of the true solution. The order of convergence is proved to be $$p+1$$ . We use our gradient recovery result to develop a robust recovery-type a posteriori error estimator for the gradient approximation which is based on an enhanced recovery technique. We prove that the proposed residual-type and recovery-type a posteriori error estimates converge to the true errors in the $$L^2$$ -norm under mesh refinement. The order of convergence is proved to be $$p + 1$$ . Moreover, the proposed estimators are proved to be asymptotically exact. Finally, we present a local adaptive mesh refinement procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for $$P^p$$ polynomials with $$p\ge 1$$ . We provide several numerical examples illustrating the effectiveness of our procedures.

Optimal superconvergence and asymptotically exact a posteriori error estimator for the local discontinuous Galerkin method for linear elliptic problems on Cartesian grids

The purpose of this paper is twofold: to study the superconvergence properties and to present an efficient and reliable a posteriori error estimator for the local discontinuous Galerkin (LDG) method for linear second-order elliptic problems on Cartesian grids. We prove that the LDG solution is superconvergent towards a particular projection of the exact solution. The order of convergence is proved to be p+2, when tensor product polynomials of degree at most p are used. Then, we prove that the actual error can be split into two parts. The components of the significant part can be given in terms of (p+1)-degree Radau polynomials. We use these results to construct a reliable and efficient residual-type a posteriori error estimates. We prove that the proposed residual-type a posteriori error estimates converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p+2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. We provide several numerical examples illustrating the effectiveness of our procedures.

Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods

Abstract We discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from 𝒪 ⁢ ( k 3 ⁢ d ) {\mathcal{O}(k^{3d})} to 𝒪 ⁢ ( d ⁢ k d + 1 ) {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d.

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

ABSTRACTIn this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.

Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems

We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order k+1 are obtained for the semi-discrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree k are used. A high-order energy-conserving Lax-Wendroff time discretization is also presented.Extensive numerical results in one dimension, and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods. One particular method (with the doubling of unknowns) is found to be optimally convergent on triangular meshes for all the examples considered in this paper. The method is also compared with the classical (dissipative) upwinding DG method and (conservative) DG method with a central flux. It is numerically observed for the new method to have a superior performance for long time simulations.

A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids

In this article, we develop and analyze a new recovery‐based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the ‐norm of the solution is of order , when tensor product polynomials of degree at most are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be . We use our derivative recovery result to develop a robust recovery‐type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery‐type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery‐type a posteriori error estimates, at a fixed time, converge to the true errors in the ‐norm under mesh refinement. The order of convergence is proved to be . Our theoretical results are valid for piecewise polynomials of degree and under the condition that each component, , of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery‐based a posteriori error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1224–1265, 2017

Superconvergence of the local discontinuous Galerkin method for the sine-Gordon equation on Cartesian grids

In this paper, we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We perform a local error analysis and show that the actual error can be split into an O(hp+1) leading component and a higher-order component, when tensor product polynomials of degree at most p are used. We further prove that the leading term of the LDG error is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. Thus, the LDG solution is O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Computational results indicate that our superconvergence results hold globally. We use these results to construct simple, efficient, and asymptotically exact a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

In this paper, we provide the first a posteriori error analysis of the discontinuous Galerkin (DG) method for solving the two-dimensional linear hyperbolic conservation laws on Cartesian grids. The key ingredients in our error analysis are the recent optimal superconvergence results proved in Cao et al. (SIAM J Numer Anal 53:1651---1671, 2015). We first prove that the DG solution converges in the $$L^2$$L2-norm to a Radau interpolating polynomial under mesh refinement. The order of convergence is proved to be $$p+2$$p+2, when tensor product polynomials of degree at most p are used. Then we show that the actual error can be divided into a significant part and a less significant part. The significant part of the DG error is spanned by two $$(p+1)$$(p+1)-degree right Radau polynomials in the x and y directions. The less significant part converges to zero at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2. These results are used to construct simple, efficient and asymptotically exact a posteriori error estimates. Superconvergence towards the right Radau interpolating polynomial is used to prove that, for smooth solutions, our a posteriori DG error estimates converge at a fixed time to the true spatial errors in the $$L^2$$L2-norm at $${\mathcal {O}}\left( h^{p+2}\right) $$Ohp+2 rate. Finally, we prove that the global effectivity indices in the $$L^2$$L2-norm converge to unity at $${\mathcal {O}}(h)$$O(h) rate. Our proofs are valid for arbitrary regular Cartesian meshes using tensor product polynomials of degree at most p and for both the periodic and Dirichlet boundary conditions. Several numerical experiments are performed to validate the theoretical results.

Superconvergence and a posteriori error estimates of the DG method for scalar hyperbolic problems on Cartesian grids

In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O(hp+1) superconvergent approximation to the directional derivative. Moreover, we establish an O(h2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O(hp+1) rate and that the global effectivity index converges to unity at O(h) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations

Abstract. In this paper, we present the optimal L2-error estimate ofO(hk+1) for polynomial elements of degree k of the semidiscrete direct discontinuous Galerkin method for convection-diffusion equations. The main technical difficulty lies in the control of the inter-element jump terms which arise because of the convection and the discontinuous nature of numerical solutions. The main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled. In multi-dimensional case, the orders of k + 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on Cartesian grids. A collection of projection errors in both oneand multi-dimensional cases is established.

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

In this paper, we propose and analyze a new superconvergent local discontinuous Galerkin (LDG) method equipped with an element residual error estimator for the spatial discretization of the second-order wave equation on Cartesian grids. We prove the L2 stability, the energy conserving property, and optimal L2 error estimates for the semi-discrete formulation. In particular, we identify special numerical fluxes for which theL2-norm of the solution and its gradient are of order p+1, when tensor product polynomials of degree at most p are used. We further perform a local error analysis and show that the leading term of the LDG error is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, the LDG solution is O(hp+2) superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree right Radau polynomial. Furthermore, numerical computations show that the first component of the solution’s gradient is O(hp+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(hp+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Computational results indicate that global superconvergence holds for LDG solutions. We use the superconvergence results to construct a posteriori LDG error estimates. These error estimates are computationally simple and are obtained by solving local steady problems with no boundary condition on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.