A posteriorierror estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems

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 second order linear PDEs. Our residual type upper and lower bound error estimates measure the error in the energy norm. 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 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. As a side product of our formulation, the penalty parameter in the interior penalty formulation can be automatically determined as well. We develop an efficient numerical procedure to compute the error estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.esaim

To improve the computational accuracy and efficiency of long-time wavefield simulations, we have developed a so-called symplectic interior penalty discontinuous Galerkin (IPDG) method for 2D acoustic equation. For the symplectic IPDG method, the scalar wave equation is first transformed into a Hamiltonian system. Then, the high-order IPDG formulations are introduced for spatial discretization because of their high accuracy and ease of dealing with computational domains with complex boundaries. The time integration is performed using an explicit third-order symplectic partitioned Runge-Kutta scheme so that it preserves the Hamiltonian structure of the wave equation in long-term simulations. Consequently, the symplectic IPDG method combines the advantages of discontinuous Galerkin method and the symplectic time integration. We investigate the properties of the method in detail for high-order spatial basis functions, including the stability criteria, numerical dispersion and dissipation relationships, and numerical errors. The analyses indicate that the symplectic SIPG method is nondissipative and retains low numerical dispersion. We also find that different symplectic IPDG methods have different convergence behaviors. It is indicated that using coarse meshes with a high-order method produces smaller errors and retains high accuracy. We have applied our method to simulate the scalar wavefields for different models, including layered models, a rough topography model, and the Marmousi model. The numerical results show that the symplectic IPDG method can suppress numerical dispersion effectively and provide accurate information on the wavefields. We also conduct a long-term experiment that verifies the capability of symplectic IPDG method for long-time simulations.

In this work, we apply the hypercircle method to Discontinuous Galerkin (DG) approximations of second order diffusion problems featuring inhomogeneous Dirichlet and Neumann boundary conditions. We focus on the interior penalty discontinuous Galerkin (IPDG) approximations of diffusion problems in primal variational formulation and produce a Prager–Synge theorem for such DG methods. Using the hypercircle method, we derive an a posteriori error estimator in terms of an equilibrated flux. The estimator is proven to be reliable and efficient. Numerical results are presented which illustrate the estimator’s performance.

This study extends the discontinuous Galerkin (DG) methods to simulations of thermoelasticity. A thermoelastic formulation of interior penalty DG (IP-DG) method is presented and aspects of the numerical implementation are discussed in matrix form. The content related to thermal expansion effects is illustrated explicitly in the discretized equation system. The feasibility of the method for general thermoelastic simulations is validated through typical test cases, including tackling stress discontinuities caused by jumps of thermal expansive properties and controlling accompanied non-physical oscillations through adjusting the magnitude of IP term. The developed simulation platform upon the method is applied to the engineering analysis of thermoelastic performance for a turbine vane and a series of vanes with various types of simplified thermal barrier coating (TBC) systems. This analysis demonstrates that while TBC properties on heat conduction are generally the major consideration for protecting the alloy base vanes, the mechanical properties may have more significant effects on protections of coatings themselves. Changing characteristics of normal tractions on TBC/base interface, closely related to the occurrence of coating failures, over diverse components distributions along TBC thickness of the functional graded materials are summarized and analysed, illustrating the opposite tendencies in situations with different thermal-stress-free temperatures for coatings.

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, includingh-refinement andp-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to theL2norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in theL2norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

Two-grid IPDG discretization scheme for nonlinear elliptic PDEs

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. 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 eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.

In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mesh refinement in the Cartesian mesh structure for interface problems. Numerical results are provided to demonstrate the convergence and local mesh refinement features of these DG-IFE methods.

A reconstructed discontinuous Galerkin (RDG) method based on a Hierarchical WENO reconstruction, termed HWENO(P1P2) in this paper, is developed for computing shock waves on 3D hybrid grids. The HWENO(P1P2) method is designed not only to enhance the accuracy of the discontinuous Galerkin (DG) method but also to ensure the nonlinear stability of the RDG method. In the HWENO(P1P2) method, a quadratic polynomial solution (P2) is first reconstructed using a WENO reconstruction from the underlying linear polynomial (P1) discontinuous Galerkin solution to ensure the linear stability of the RDG method and to improve the efficiency of the underlying DG method. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities and thus ensure the nonlinear stability of the RDG method. The developed HWENO(P1P2) method is used to compute a variety of shock wave problems on 3D hybrid meshes to demonstrate its accuracy, robustness, versatility, and essentially non-oscillatory property. The numerical experiments indicate that the HWENO(P1P2) method is able to provide sharp resolution of discontinuities essentially without any spurious oscillations, and achieve the designed third-order of accuracy for smooth flows.

On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method

In this report we present a new approach to simulations on complex shaped domains. The method uses a Discontinuous Galerkin discretization and a structured grid to construct the test and trial functions. Boundary and transmission conditions along the complex shape of the domains are imposed weakly via the Discontinuous Galerkin formulation. This method o ers a discretization where the minimal number of unknowns is independent of the possibly very complicated shape of the domain.

Orthogonal time frequency space (OTFS) modulations are robust to time-varying frequency-selective fading channels. OTFS modulations operate in the delay-Doppler domain whereas two-dimensional (2D) orthogonal frequency division multiplexing (OFDM) modulations operate in the time-frequency domain. For 2D OFDM modulations in time-varying frequency-selective fading channels, we investigate the pilot pattern design problem, which minimizes the mean square error (MSE) of channel estimation. The MSE lower bound (LB) is theoretically derived to provide the design criterion. Based on the criterion, we show that the LB achieving design can be found by exhaustive 2D pilot location search. Exhaustive 2D search has high computational complexity. To reduce the complexity, sufficient conditions on the existence of LB achieving design are provided to decouple the 2D problem into two one-dimensional (1D) problems. For the 1D problem, we propose the LB achieving design when the number of possible pilot locations is divisible by the number of pilots. Simulation results illustrate that our proposed LB achieving design is superior to the random pilot pattern design and the bilinear interpolation channel estimation method.