A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

Abstract

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests

A Multilevel Algebraic Error Estimator and the Corresponding Iterative Solver with $p$-Robust Behavior

“…An inferior quality of the contraction factors for the case of p = 1 and the use of damping factors w 1 = J(d + 1) and w 2 = 1 appears. This is in line with some precedents in literature, where numerically probust solvers also perform worse for order 1 approximations; we mention, for example, Griebel, Oswald, and √ J, w2 = ∞ w1 = d + 1, w2 = J "small" "large" "small" "large" "small" "large" J p DoF isᾱ isᾱ isᾱ isᾱ isᾱ isᾱ 3 1 5e 3 48 .3), p = p in (2.10a), "small" and "large" patches. i s : the number of iterations needed to reach the stopping criterion (6.4).ᾱ: average error contraction factor given by (6.5).…”

“…This gives a spectacular gain in the number of iterations, and makes the method convergent even when the standard Jacobi fails. "large" ν = 1 "small" "large" ν = 1 "small" "large" ν = 1 J p ν = 1 ν = 3 ν = 1 ν = 3 J GS ν = 1 ν = 3 ν = 1 ν = 3 J GS ν = 1 ν = 3 ν = 1 ν = 3 J GS 3 .3), p = p in (2.10a), "small" and "large" patches, ν postsmoothing steps, and standard multigrid method with piecewise affine coarse solve (3.3), initialized by the coarse grid solution, no presmoothing, one postsmoothing step, and Jacobi (J) and Gauss-Seidel (GS) smoothers.…”

“…More recent works were carried out based on these approaches. In Antonietti et al [3] (see also the references therein), the p-robust approach for rectangular/hexahedral meshes was used for high-order discontinuous Galerkin (DG) methods; moreover the spectral bounds of the preconditioner are also robust with respect to the method's penalization coefficient. We also mention the introduction of multilevel preconditioners yielded by block Gauss-Seidel smoothers in Kanschat [29] for rectangular/hexahedral meshes and DG discretizations.…”

Multigrid algorithms for $$\varvec{hp}$$ h p -version interior penalty discontinuous Galerkin methods on polygonal and polyhedral meshes

“…For the sake of comparison, we also report the iteration counts N PCG it for the Preconditioned Conjugate Gradient (PCG) method, based on employing a simple block Jacobi preconditioner. The extension to polytopic grids of the domain decomposition preconditioning techniques, such as, for example, the ones proposed in [9,11,14], in the DG setting, or in [51,54], in the conforming setting, are currently under investigation and will be the subject of future research. Table 3 presents analogous results for the first three sets of meshes, in the case when p = 3.…”

On a nonoverlapping additive Schwarz method for h‐p discontinuous Galerkin discretization of elliptic problems

“…In addition to , other results related to preconditioning discontinuous Galerkin discretizations of elliptic PDEs include a BDDC preconditioner and a substructuring method . Recently, an additive Schwarz method utilizing conforming subspaces of order has been introduced in and proved optimal in both and , in the case of constant . While these papers consider preconditioners which provide better dependence on the polynomial order, the algorithm analyzed here is very simple to implement, puts less restrictions on the conformity of elements and meshes, and provides two additional levels of customization: the choice of the order of the coarse grid approximation and the granularity of the parallelism (by the choice of the size of subdomains).…”