We review the derivation of weakly penalized discontinuous Galerkin methods for scattering dominated radiation transport and extend the asymptotic analysis to non-isotropic scattering.
We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.
We consider a discontinuous interpolation operator with a parameter that controls the discontinuity, and determine the optimal choice for the discontinuity, leading to the fastest solver for a specific 1D symmetric interior penalty DG discretization model problem. We show in addition that our optimization delivers a perfectly clustered spectrum with a high geometric multiplicity, which is very advantageous for a Krylov solver using the method as its preconditioner.
We present an efficient solver for the linear system that arises from the Hierarchical Poincaré-Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. Numerical results show the capacity to tackle problems of approximately 100 wavelengths in each direction, requiring more than a billion unknowns to achieve approximately 4 digits of accuracy taking less than 20 minutes to solve.
We apply two-level methods to a symmetric interior penalty discontinuous Galerkin finite element discretization of a singularly perturbed reactions-diffusion equation. The main innovation is that explicit formulas for the optimal relaxation parameter of the two-level method for the Poisson problem in 1D are obtained, as well as very accurate closed form approximation formulas for the optimal choice in the reaction-diffusion case in all regimes. Numerical experiments and comparisons show the applicability of the expressions obtained in higher dimensions and general geometries.
We revisit the Hierarchical Poincaré–Steklov (HPS) method for the Poisson equation using standard Q1 finite elements, building on the original in [1]. While corner degrees of freedom were implicitly handled in that work, subsequent spectral-element implementations have typically avoided them. In Q1-FEM, however, corner coupling cannot be factored out, and we show how the HPS merge procedure naturally accommodates it when corners are enclosed by elements. This clarification bridges a conceptual gap between algebraic Schur-complement methods and operator-based formulations, providing a consistent path for the FEM community to adopt HPS to retain the Poincaré–Steklov interpretation at both continuous and discrete levels.
As part of the development of a Poisson solver for the spectral element discretization used in the GeoFluid Object Workbench (GeoFLOW) code, we propose a solver for the linear system arising from a Gauss-Legendre-Lobatto global spectral method. We precondition using a p-multigrid γ-cycle with highly-vectorizable smoothers, that we refer to as line smoothers. Our smoothers are restrictions of spectral and finite element discretizations to low-order one-dimensional problems along lines, that are solved by a reformulation of cyclic reduction as a direct multigrid method. We illustrate our method with numerical experiments showing the apparent boundedness of the iteration count for a fixed residual reduction over a range of moderately deformed domains, right hand sides and Dirichlet boundary conditions.
We revisit the Hierarchical Poincaré–Steklov (HPS) method within a preconditioned iterative framework. Originally introduced as a direct solver for elliptic boundary-value problems, the HPS method combines nested dissection with tensor-product spectral element discretizations, even though it has been shown in other contexts [1]. Building on the iterative variant proposed in [2], we reinterpret the hierarchical merge structure of HPS as a natural multigrid preconditioner. This perspective unifies direct and iterative formulations of HPS connecting it to multigrid domain decomposition. The resulting formulation preserves the high accuracy of spectral discretizations while enabling flexible iterative solution strategies. Numerical experiments in two dimensions demonstrate the performance and convergence behavior of the proposed approach.
This paper studies a common two-level multigrid construction for block-structured linear systems and identifies a simple way to describe how its smoothing and coarse-grid components interact. By examining the method through a collection of small coupling modes, we show that its behavior can be captured by a single scalar quantity for each mode. The main result is an explicit choice of smoothing parameters that makes all modes respond in the same way, causing the nontrivial eigenvalues of the preconditioned operator to collapse to a single value. This gives a clear and self-contained description of the ideal version of the method and provides a concrete target for designing related schemes. Although the exact spectral collapse requires ideal components, we also show that the same construction naturally produces operators that resemble those used in practical discretizations. Examples from finite-difference and discontinuous Galerkin settings illustrate how the ideal parameters can be used in practice.