Adaptive Discontinuous Galerkin Methods for Eigenvalue Problems Arising in Incompressible Fluid Flows

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z 2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual–Weighted–Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three‐dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable , we rewrite the problem into a two‐order system. Then, the a priori error estimates are derived in L2 norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r () are used, we obtain the optimal convergence rate of order r + 1 in L2 norm and of order r in DG norm for u, and the order r in both norms for . The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017

We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.

For elliptic interface problems in two and three dimensions, this paper studies a priori and residual-based a posteriori error estimations for the Crouzeix--Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with no assumption on the distribution of the diffusion coefficient.

In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on ...

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

In this work the numerical discretization of the partial differential governing equations for compressible and incompressible flows is dealt within the discontinuous Galerkin (DG) framework along space–time adaptive meshes. Two main numerical frameworks can be distinguished: (1) fully explicit ADER-DG methods on collocated grids for compressible fluids (2) spectral semi-implicit and spectral space–time DG methods on edge-based staggered grids for the incompressible Navier–Stokes equations. In this work, the high-resolution properties of the aforementioned numerical methods are significantly enhanced within a 'cell-by-cell' Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called 'Gibbs phenomenon'. Over the years, several attempts have been made to cope with this problem and different kinds of limiters have been proposed. In this work the nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a multidimensional optimal order detection (MOOD) criterion. ADER-DG is a novel, communication-avoiding family of algorithms, which achieves high order of accuracy in time not via the standard multi-stage Runge–Kutta (RK) time discretization like most other DG schemes, but at the aid of an element-local predictor stage. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, in those cells where at least one of the chosen admissibility criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Next, the numerical solution of the previous time step is scattered onto cell averages on a suitable sub-grid in order to preserve the natural sub-cell resolution of the DG scheme. Then, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the sub-grid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved sub-cell averages. In the ADER-DG framework several PDE system are investigated, ranging from the Euler equations of compressible gas dynamics, over the viscous and resistive magneto-hydrodynamics (MHD), to special and general relativistic MHD. Indeed, the adopted formalism is quite general, leading to a novel family of adaptive ADER-DG schemes suitable for hyperbolic systems of partial differential equations in which the numerical fluxes also depend on the gradient of the state vector because of the parabolic nature of diffusive terms. The presented results show clearly that the high-resolution and shock-capturing capabilities of the news schemes are significantly enhanced within the cell-by-cell AMR implementation together with time accurate LTS. A special treatment has been followed for the incompressible Navier–Stokes equations. In fact, the elliptic character of the incompressible Navier–Stokes equations introduces an important difficulty in their numerical solution: whenever the smallest physical or numerical perturbation arises in the fluid flow then it will instantaneously affect the entire computational domain. Thus, a semi-implicit approach has been used. The main advantage of making use of a semi-implicit discretization is that the numerical stability can be obtained for large time-steps without leading to an excessive computational demand. In this context, we derived two new families of spectral semi-implicit and spectral space–time DG methods for the solution of the two and three dimensional Navier–Stokes equations on edge-based adaptive staggered Cartesian grids. The discrete solutions of pressure and velocity are expressed in the form of piecewise polynomials along different meshes. While the pressure is defined on the control volumes of the main grid, the velocity components are defined on edge-based dual control volumes, leading to a spatially staggered mesh. In the first family, high order of accuracy is achieved only in space, while a simple semi-implicit time discretization is derived by introducing an implicitness factor $$\theta \in [0.5,1]$$ for the pressure gradient in the momentum equation. The real advantages of the staggering arise after substituting the discrete momentum equation into the weak form of the continuity equation. In fact, the resulting linear system for the pressure is symmetric and positive definite and either block penta-diagonal (in 2D) or block hepta-diagonal (in 3D). As a consequence, the pressure system can be solved very efficiently by means of a classical matrix-free conjugate gradient method. The resulting algorithm is stable, computationally very efficient, and at the same time arbitrary high order accurate in both space and time. This new numerical method has been thoroughly validated for approximation polynomials of degree up to $$N = 12$$, using a large set of non-trivial test problems in two and three space dimensions, for which either analytical, numerical or experimental reference solutions exist.

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second order linear operators. Eigenvalue problems of such types play important roles in scientific and engineering applications, particularly in theoretical chemistry, solid state physics and material science. Based on the framework developed in [{\it L. Lin, B. Stamm, http://dx.doi.org/10.1051/m2an/2015069}] for second order PDEs, we develop residual type upper and lower bound error estimates for measuring the a posteriori error for eigenvalue problems. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local and independent eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. Compared to the PDE case, we find that a posteriori error estimators for eigenvalue problems must neglect certain terms, which involves explicitly the exact eigenvalues or eigenfunctions that are not accessible in numerical simulations. We define such terms carefully, and justify numerically that the neglected terms are indeed numerically high order terms compared to the computable estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective for measuring the error of eigenvalues and eigenfunctions.

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

This letter proposes to automate generation of reduced-order models used for accelerated S-parameter computation by applying a posteriori model error estimators. So far, a posteriori error estimators were used in Reduced Basis Method (RBM) and Proper Orthogonal Decomposition (POD) to select frequency points at which basis vectors are generated. This letter shows how a posteriori error estimators can be applied to automatically select the order of the reduced model in second-order Model Order Reduction (MOR) methods. Three different error estimators are investigated and compared in order to arrive at a new MOR scheme that is fast, reliable, and fully automated. The effectiveness of the proposed approach is verified by very high accuracy of the computed scattering parameters ( S-parameters) for an example of a waveguide filter over a prescribed frequency band.

We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuska-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

This is a further development of [9] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either continuous or piecewise constant functions. The spaces are nested, but due to the penalty term in the DG method the corresponding forms are not inherited. For the continuous finite element spaces we show that the variable V-cycle provides an optimal preconditioner for the DG system. The piece-wise constant functions do not have approximation property so in order to control the energy growth of the inter-level transfer operator we apply W–cycle MG. Finally, we present a number of numerical experiments that support the theoretical findings.

It is shown that the interelement discontinuities in a discontinuous Galerkin finite element approximation are subordinate to the error measured in the broken $H^1$-seminorm. One consequence is that the DG-norm of the error is equivalent to the broken energy seminorm. Computable a posteriori error bounds are obtained for the error measured in both the DG-norm and the broken energy seminorm and are shown to be efficient in the sense that they also provide lower bounds up to a constant and higher order data oscillation terms. The estimators are completely free of unknown constants and provide guaranteed numerical bounds for the error.

We construct and analyze a numerical scheme for the two-dimensional Vlasov-Poisson system based on a backward-Euler (BE) approximation in time combined with a mixed finite element method for a discretization of the Poisson equation in the spatial domain and a discontinuous Galerkin (DG) finite element approximation in the phase-space variables for the Vlasov equation. We prove the stability estimates and derive the optimal convergence rates depending upon the compatibility of the finite element meshes, used for the discretizations of the spatial variable in Poisson (mixed) and Vlasov (DG) equations, respectively. The error estimates for the Poisson equation are based on using Brezzi-Douglas-Marini (BDM) elements in L2 and H?s, s>0, norms.