Abstract. We present a scalable iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of linear partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations, and hence inheriting advances from both sides. In particular, the method can be viewed as a Gauss-Seidel approach that requires only independent element-by-element and face-by-face local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. Unlike conventional Gauss-Seidel schemes which are purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical flux, does not depend on the ordering of unknowns. We rigorously show the convergence of the proposed method for the transport equation, the linearized shallow water equation and the convectiondiffusion equation. For the transport equation, the method is convergent regardless of mesh size h and solution order p, and furthermore the convergence rate is independent of the solution order. For the linearized shallow water and the convection-diffusion equations we show that the convergence is conditional on both h and p. Extensive steady and time-dependent numerical results for the 2D and 3D transport equations, the linearized shallow water equation, and the convection-diffusion equation are presented to verify the theoretical findings.

We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h−coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends only on the smoothness of the solution and is independent of the nature of the PDE under consideration. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms.

No abstract

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable to any locally conservative hybridized finite element method including multinumerics with different hybridized methods in different parts of the domain. For these methods, the linear system involves only the unknowns residing on the mesh skeleton, and constructing intergrid transfer operators is therefore not trivial. The key to our geometric multigrid algorithm is the physics-based energy-preserving intergrid transfer operators which depend only on the fine scale DtN maps. Thanks to these operators, we completely avoid upscaling of parameters and no information regarding subgrid physics is explicitly required on coarse meshes. Moreover, our algorithm is agglomeration-based and can straightforwardly handle unstructured meshes. We perform extensive numerical studies with hybridized mixed methods, hybridized discontinuous Galerkin method, weak Galerkin method, and a hybridized version of interior penalty discontinuous Galerkin methods on a range of elliptic problems including subsurface flow through highly heterogeneous porous media. We compare the performance of different smoothers and analyze the effect of stabilization parameters on the scalability of the multigrid algorithm.

There is great need for high intensity proton beams from compact particle accelerators in particle physics, medical isotope production, and materials- and energy-research. To address this need, we present, for the first time, a design for a compact isochronous cyclotron that will be able to deliver 10 mA of 60 MeV protons - an order of magnitude higher than on-market compact cyclotrons and a factor four higher than research machines. A key breakthrough is that vortex motion is incorporated in the design of a cyclotron, leading to clean extraction. Beam losses on the septa of the electrostatic extraction channels stay below 120~W (40\% below the required safety limit), while maintaining good beam quality. We present a set of highly accurate particle-in-cell simulations, and an uncertainty quantification of select beam input parameters using machine learning, showing the robustness of the design. This design can be utilized for beams for experiments in particle and nuclear physics, materials science and medical physics as well as for industrial applications.