Discontinuous Petrov-Galerkin method based on the optimal test space norm for one-dimensional transport problems

We revisit the finite element analysis of convection dominated flow problems within the recently developed Discontinuous Petrov-Galerkin (DPG) variational framework. We demonstrate how test function spaces that guarantee numerical stability can be computed automatically with respect to the so called optimal test space norm by using an element subgrid discretization. This should make the DPG method not only stable but also robust, that is, uniformly stable with respect to the P'eclet number in the current application. The effectiveness of the algorithm is demonstrated on two problems for the linear advection-diffusion equation. © 2011 Published by Elsevier Ltd

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov–Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. A concise proposal on how to overcome these shortcomings has been raised during the last decade by the Discontinuous Petrov–Galerkin (DPG) methodology. However, DPG has also some limitations and difficulties: the method requires ultraweak variational formulations, obtained through a hybridization process, which is not trivial to implement at the discrete level. Our motivation is to offer a simpler alternative for the case of parametric PDEs, which can be used with any variational formulation. Indeed, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently in an online stage, for a given range of parameters. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE parameter, to the matrix of coefficients of optimal test functions (in some basis expansion) associated with that PDE parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE parameters). When solving online the resulting (compressed) Petrov–Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix–vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure is as fast as an (unstable) Galerkin approach. We illustrate our findings by means of 2D–3D Eriksson–Johnson problems, together with 2D Helmholtz equation

Various applications of discontinuous Petrov-Galerkin (DPG) finite element methods

Discontinuous Petrov-Galerkin (DPG) finite element methods have garnered significant attention since they were originally introduced. They discretize variational formulations with broken (discontinuous) test spaces and are crafted to be numerically stable by implicitly computing a near-optimal discrete test space as a function of a discrete trial space. Moreover, they are completely general in the sense that they can be applied to a variety of variational formulations, including non-conventional ones that involve non-symmetric functional settings, such as ultraweak variational formulations. In most cases, these properties have been harnessed to develop numerical methods that provide robust control of relevant equation parameters, like in convection-diffusion problems and other singularly perturbed problems. In this work, other features of DPG methods are systematically exploited and applied to different problems. More specifically, the versatility of DPG methods is elucidated by utilizing the underlying methodology to discretize four distinct variational formulations of the equations of linear elasticity. By taking advantage of interface variables inherent to DPG discretizations, an approach to coupling different variational formulations within the same domain is described and used to solve interesting problems. Moreover, the convenient algebraic structure in DPG methods is harnessed to develop a new family of numerical methods called discrete least-squares (DLS) finite element methods. These involve solving, with improved conditioning properties, a discrete least-squares problem associated with an overdetermined rectangular system of equations, instead of directly solving the usual square systems. Their utility is demonstrated with illustrative examples. Additionally, high-order polygonal DPG (PolyDPG) methods are devised by using the intrinsic discontinuities present in ultraweak formulations. The resulting methods can handle heavily distorted non-convex polygonal elements and discontinuous material properties. A polygonal adaptive strategy was also proposed and compared with standard techniques. Lastly, the natural high-order residual-based a posteriori error estimator ingrained within DPG methods was further applied to problems of physical relevance, like the validation of dynamic mechanical analysis (DMA) calibration experiments of viscoelastic materials, and the modeling of form-wound medium-voltage stator coils sitting inside large electric machinery.Computational Science, Engineering, and Mathematic

Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics

Initial mesh design for computational fluid dynamics can be a time-consuming and expensive process. The stability properties and nonlinear convergence of most numerical methods rely on a minimum level of mesh resolution. This means that unless the initial computational mesh is fine enough, convergence can not be guaranteed. Any meshes below this minimum resolution level are termed to be in the ``pre-asymptotic regime.'' This condition implies that meshes need to in some way anticipate the solution before it is known. On top of the minimum requirement that the surface meshes must adequately represent the geometry of the problem under consideration, resolution requirements on the volume mesh make the CFD practitioner's job significantly more time consuming. In contrast to most other numerical methods, the discontinuous Petrov-Galerkin finite element method retains exceptional stability on extremely coarse meshes. DPG is also inherently very adaptive. It is possible to compute the residual error without knowledge of the exact solution, which can be used to robustly drive adaptivity. This results in a very automated technology, as the user can initialize a computation on the coarsest mesh which adequately represents the geometry then step back and let the program solve and adapt iteratively until it resolves the solution features. A common complaint of minimum residual methods by computational fluid dynamics practitioners is that they are not locally conservative. In this thesis, this concern is addressed by developing a locally conservative DPG formulation by augmenting the system with Lagrange multipliers. The resulting DPG formulation is then proved to be robust and shown to produce superior numerical results over standard DPG on a selection of test problems. Adaptive convergence to steady incompressible and compressible Navier-Stokes solutions was explored in Jesse Chan's and Nathan Roberts' dissertations. Space-time offers a natural extension to transient problems as it preserves the stability and adaptivity properties of DPG in the time dimension. Space-time also offers more extensive parallelization capability than problems treated with traditional time stepping as it allows multigrid concurrently in both space and time. A proof of concept space-time DPG formulation is developed for transient convection-diffusion. The robust test norms derived for steady convection-diffusion are extended to the space-time case and proofs of robustness are provided. Numerical results verify the robust behavior and near $L^2$ optimality of the resulting solutions. The space-time formulation for convection-diffusion is then extended to transient incompressible and compressible Navier-Stokes by analogy. Several numerical experiments are performed, but a mathematical analysis is not attempted for these nonlinear problems. Several side topics are explored such as a study of the compressible Navier-Stokes equations under various variable transformations and the development of consistent test norms through the concept of physical entropy.Computational Science, Engineering, and Mathematic

Computational Engineering

The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods