Adaptive Discontinuous Galerkin Method Research Articles

Convergence of an adaptive C0-interior penalty Galerkin method for the biharmonic problem

Abstract We develop a basic convergence analysis for an adaptive ${C}^{0}\mathsf{IPG}$ method for the biharmonic problem that provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the previous convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, here we have to deal with the fact that the Lagrange finite element spaces may possibly contain no proper $C^{1}$-conforming subspace. This prevents from a straight forward generalization and requires the development of some new key technical tools.

A p-adaptive discontinuous Galerkin method for solving second-order seismic wave equations

A <i>p</i>-adaptive discontinuous Galerkin method for solving second-order seismic wave equations

A parallelp‐adaptive discontinuous Galerkin method for the Euler equations with dynamicload‐balancingon tetrahedral grids

AbstractA novelp‐adaptive discontinuous Galerkin (DG) method has been developed to solve the Euler equations on three‐dimensional tetrahedral grids. Hierarchical orthogonal basis functions are adopted for the DG spatial discretization while a third order TVD Runge‐Kutta method is used for the time integration. A vertex‐based limiter is applied to the numerical solution in order to eliminate oscillations in the high order method. An error indicator constructed from the solution of order and is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. The developed method is implemented with under the Charm++ parallel computing framework. Charm++ is a parallel computing framework that includes various load‐balancing strategies. Implementing the numerical solver under Charm++ system provides us with access to a suite of dynamic load balancing strategies. This can be efficiently used to alleviate the load imbalances created byp‐adaptation. A number of numerical experiments are performed to demonstrate both the numerical accuracy and parallel performance of the developedp‐adaptive DG method. It is observed that the unbalanced load distribution caused by the parallelp‐adaptive DG method can be alleviated by the dynamic load balancing from Charm++ system. Due to this, high performance gain can be achieved. For the testcases studied in the current work, the parallel performance gain ranged from 1.5× to 3.7×. Therefore, the developedp‐adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method withoutp‐adaptation.

A space-time adaptive discontinuous Galerkin method for the numerical solution of the dynamic quasi-static von Kármán equations

A space-time adaptive discontinuous Galerkin method for the numerical solution of the dynamic quasi-static von Kármán equations

Geometrically consistent aerodynamic optimization using an isogeometric Discontinuous Galerkin method

The objective of the current work is to define a design optimization methodology in aerodynamics, in which all numerical components are based on a unique geometrical representation, consistent with Computer-Aided Design (CAD) standards. In particular, the design is parameterized by Non-Uniform Rational B-Splines (NURBS), the computational domain is automatically constructed using rational Bézier elements extracted from NURBS boundaries without any approximation and the resolution of the flow equations relies on an adaptive Discontinuous Galerkin (DG) method based on rational representations. A Bayesian framework is used to optimize NURBS control points, in a single- or multi-objective, constrained, global optimization framework. The resulting methodology is therefore fully CAD-consistent, high-order in space and time, includes local adaption and shock capturing capabilities, and exhibits high parallelization performance. The proposed methods are described in details and their properties are established. Finally, two design optimization problems are provided as illustrations: the shape optimization of an airfoil in transonic regime, for drag reduction with lift constraint, and the multi-objective optimization of the control law of a morphing airfoil in subsonic regime, regarding the time-averaged lift, the minimum instantaneous lift and the energy consumption.

An adaptive discontinuous Galerkin method for the Darcy system in fractured porous media

An adaptive discontinuous Galerkin method for the Darcy system in fractured porous media

Adaptive local discontinuous Galerkin methods with semi-implicit time discretizations for the Navier-Stokes equations

In this paper, we present a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations under the framework of local discontinuous Galerkin methods coupled with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods. In both of the two high order semi-implicit time integration methods, the convective flux is treated explicitly and the viscous and heat fluxes are treated implicitly. The remarkable benefits of such semi-implicit temporal discretizations are that they can not only overcome the stringent time step restriction compared with time explicit methods, but also avoid the construction of the large Jacobian matrix as is done for fully implicit methods, thus are relatively easy to implement. To save computing time as well as capture the flow structures of interest accurately, a local mesh refinement (h-adaptive) technique, in which we present detailed criteria for selecting candidate elements and complete strategies to refine and coarsen them, is also applied for the Navier-Stokes equations. Numerical experiments are provided to illustrate the high order accuracy, efficiency and capabilities of the semi-implicit schemes in combination with adaptive local discontinuous Galerkin methods for the Navier-Stokes equations.

A local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations

We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions (Abdulle and Rosilho de Souza, 2019) [1]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy. Reliable error estimators are derived for the local adaptive method. Numerical comparisons with a classical nonlocal adaptive algorithm illustrate the efficiency of the method.

Spatio-stochastic adaptive discontinuous Galerkin methods

We introduce a goal-oriented, adaptive framework for the uncertainty quantification of systems modeled by stochastically parametrized nonlinear hyperbolic and convection-dominated partial differential equations. Of particular interest are conservation laws in aerodynamics that may have a small number of stochastic parameters but exhibit strong nonlinearity and a wide range of scales. Our framework exploits localized structure in the spatio-parameter space to enable rapid, reliable uncertainty quantification for output quantities of interest. Our formulation comprises the following technical components: (i) a discontinuous Galerkin finite element method, which provides stability for convection-dominated problems; (ii) element-wise polynomial chaos expansions, which capture the parametric dependence of the solution in a way amenable to adaptation; (iii) the dual-weighted residual method, which provides global and element-wise error estimates for quantities of interest; and (iv) a projection-based anisotropic error indicator along with the associated adaptation mechanics that can detect and refine strongly directional features in the physical and/or parameter spaces simultaneously in an efficient manner. Both the spatial and stochastic discretization errors are controlled through the adaptive refinement of the spatial mesh or polynomial chaos expansion degree based on these anisotropic error indicators. We analyze stability, approximation properties, and a priori error bounds of the spatio-stochastic adaptive method. We finally demonstrate the effectiveness of our formulation for engineering-relevant transonic turbulent aerodynamics problems with uncertainties in flow conditions and turbulence parameters.

Discontinuous Galerkin Methods and Their Adaptivity for the Tempered Fractional (Convection) Diffusion Equations

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are respectively verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.

Entropy-Adjoint -Adaptive Discontinuous Galerkin Method for the Under-Resolved Simulation of Turbulent Flows

This paper presents an approach to mesh adaptation suitable for scale-resolving simulations. The methodology is based on the entropy-adjoint approach, which corresponds to a standard output-based adjoint method with the output functional targeting areas of spurious generation of entropy. The method shows several advantages over standard output-based error estimation: 1) it is computationally inexpensive; 2) it does not require the solution of a fine-space adjoint problem; and 3) it is nonlinearly stable with respect to the primal solution for chaotic dynamic systems. In addition, the work reports on the parallel efficiency of the solver, which has been optimized through a multiconstraint domain decomposition algorithm available within the Metis 5.0 library. The reliability, accuracy, and efficiency of the approach are assessed by computing three test cases: the two-dimensional, laminar, chaotic flow around a square at ; and the implicit large-eddy simulation of the flow past a circular cylinder at and past a square cylinder at . The results show a significant reduction in the number of degrees of freedom with respect to uniform order refinement with a good agreement with experimental data.

An adaptive discontinuous Galerkin method for the simulation of hurricane storm surge

Numerical simulations based on solving the 2D shallow water equations using a discontinuous Galerkin (DG) discretisation have evolved to be a viable tool for many geophysical applications. In the context of flood modelling, however, they have not yet been methodologically studied to a large extent. Systematic model testing is non-trivial as no comprehensive collection of numerical test cases exists to ensure the correctness of the implementation. Hence, the first part of this manuscript aims at collecting test cases from the literature that are generally useful for storm surge modellers and can be used to benchmark codes. On geographic scale, hurricane storm surge can be interpreted as a localised phenomenon making it ideally suited for adaptive mesh refinement (AMR). Past studies employing dynamic AMR have exclusively focused on nested meshes. For that reason, we have developed a DG storm surge model on a triangular and dynamically adaptive mesh. In order to increase computational efficiency, the refinement is driven by physics-based refinement indicators capturing major model sensitivities. Using idealised numerical test cases, we demonstrate the model’s ability to correctly represent all source terms and reproduce known variability of coastal flooding with respect to hurricane characteristics such as size and approach speed. Finally, the adaptive mesh significantly reduces computing time with no effect on storm waves measured at discrete wave gauges just off the coast which shows the model’s potential for use as a robust simulation tool for real-time predictions.

Adaptive discontinuous Galerkin methods for solving an incompressible Stokes flow problem with slip boundary condition of frictional type

We consider the numerical solution of a variational inequality arising in the study of an incompressible Stokes flow with a nonlinear slip boundary condition of frictional type. We study several discontinuous Galerkin (DG) schemes for solving the problem, and derive reliable and efficient residual type a posteriori error estimators. An adaptive DG method is constructed based on the error estimators to solve the problem. Some numerical examples are presented to illustrate the efficiency of the adaptive DG method.

Multi-hp adaptive discontinuous Galerkin methods for simplified PN approximations of 3D radiative transfer in non-gray media

In this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified PN approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified PN models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified PN approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified PN approximations of radiative transfer in non-gray media.

Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems

We prove a basic error contraction result of an adaptive discontinuous Galerkin method for an elliptic interface problem. The interface conditions considered model mass transfer of solutes through semi-permeable membranes and other filtering processes. The adaptive algorithm is based on a residual-type a posteriori error estimator, with a bulk refinement criterion. The a posteriori error bound is derived under the assumption that the triangulation is aligned with the interfaces although, crucially, extremely general curved element shapes are also allowed, resolving the interface geometry exactly. As a corollary, convergence of the adaptive discontinuous Galerkin method for non-essential Neumann- and/or Robin-type boundary conditions, posed on general curved boundaries, also follows. Numerical experiments are also presented.

Formula omitted]-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems

An a posteriori error estimator for the error in the (L2(H1)+L∞(L2))-type norm for an interior penalty discontinuous Galerkin (dG) spatial discretisation and backward Euler temporal discretisation of linear non-stationary convection–diffusion initial/boundary value problems is derived, allowing for anisotropic elements. The proposed error estimator is used to drive an hp-space–time adaptive algorithm wherein directional mesh refinement is employed to give rise to highly anisotropic elements able to accurately capture layers. The performance of the hp-space–time adaptive algorithm is assessed via a number of standard test problems characterised by sharp and/or moving layers.

Convergence of adaptive discontinuous Galerkin methods

Convergence of adaptive discontinuous Galerkin methods

Adaptive discontinuous Galerkin methods for elliptic interface problems

Adaptive discontinuous Galerkin methods for elliptic interface problems

An adjoint-based grid adaptive discontinuous Galerkin method

A grid adaptation method has been developed for discontinuous Galerkin discretizations of two-dimensional Euler equations. The element contribution to the global error in interested output variable is used as an indicator to drive mesh adaptation strategies. In the error indicator construction process, the adjoint variables are obtained by solving the adjoint equations with the GMRES algorithm. The solutions of the primal and adjoint problems in the enriched space of order p +1 are approximated by taking a small number of block-diagonal Jacobi iterations on the projected solutions of order p . In order to enforce the adjoint consistency, the numerical flux and output functions defined on the wall are constructed to depend only on the exterior states. The drag coefficient of NACA0012 airfoil with different orders of accuracy is used to evaluate the reliability of the output error estimation method. The subsonic and transonic inviscid flows around NACA 0012 airfoil, and the Mach 6 hypersonic flow over a half cylinder are simulated using the developed grid adaptation method. The computation results demonstrate that the adjoint-based adaptive method is very effective in reducing discretiztion errors of the output functions such as drag coefficient. The results for the subsonic flow of NACA 0012 shows that only 17% of degrees of freedom employed by the uniform refinement is required to achieve equivalent output accuracy by the adjoint-based adaptive approach.

An agglomeration‐based adaptive discontinuous Galerkin method for compressible flows

SummaryIn this work, we exploit the possibility to devise discontinuous Galerkin discretizations over polytopic grids to perform grid adaptation strategies on the basis of agglomeration coarsening of a fine grid obtained via standard unstructured mesh generators. The adaptive agglomeration process is here driven by an adjoint‐based error estimator. We investigate several strategies for converting the error field estimated solving the adjoint problem into an agglomeration factor field that is an indication of the number of elements of the fine grid that should be clustered together to form an agglomerated element. As a result the size of agglomerated elements is optimized for the achievement of the best accuracy for given grid size with respect to the target quantities. To demonstrate the potential of this strategy we consider problem‐specific outputs of interest typical of aerodynamics, eg, the lift and drag coefficients in the context of inviscid and viscous flows test cases.