Time-Parallel Iterative Solvers for Parabolic Evolution Equations

Neumüller, Martin; Smears, Iain

doi:10.1137/18m1172466

Cited by 19 publications

(11 citation statements)

References 42 publications

(55 reference statements)

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“…(The uniform stability holds true if the mesh sequence is quasi-uniform, see also Remark 22.23.) The reader is referred to Smears [262], Neumüller and Smears [229] for further results on the inf-sup stability of dG(k) schemes with a time-independent bilinear form a.…”

Section: Putting Everything Together and Recalling The Identity From Stepmentioning

confidence: 99%

Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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In Part XII, composed of Chapters 56 to 63, we study the finite element approximation of PDEs where a coercivity property is not available, so that the analysis solely relies on inf-sup conditions. Stability can be obtained by employing various stabilization techniques (residual-based or fluctuation-based). In the present chapter, we introduce the prototypical model problem we are going to work on: it is a system of first-order linear PDEs introduced in 1958 by Friedrichs [131]. This system enjoys symmetry and positivity properties and is often referred to in the literature as Friedrichs' system. Friedrichs wanted to handle within a single functional framework PDEs that are partly elliptic and partly hyperbolic, and for this purpose he developed a formalism that goes beyond the traditional classification of PDEs into elliptic, parabolic, and hyperbolic types. Friedrichs' formalism is very powerful and encompasses several model problems. Important examples are the advection-reaction equation, the div-grad problem related to Darcy's equations, and the curl-curl problem related to Maxwell's equations. This theory will be used systematically in the following chapters. All the theoretical arguments in this chapter are presented assuming that the functions are complex-valued. The real-valued case can be obtained by replacing the field C by R, by replacing the Hermitian transpose Z H by the transpose Z T , and by removing the real part symbol ℜ. Basic ideasLet D be a Lipschitz domain in R d . We consider functions defined on D with values in C m for some integer m ≥ 1. The (Hermitian) inner product inWe denote by I m the identity matrix in C m×m .M , we observe that u − I h0 (u) ∈ V 0 . Using (56.26), we infer that for all v ∈ V 0 = ker(M − N ),with φ := max(ℓ * , µ 0 h). If the boundary of D is not smooth and/or if the coefficients σ, µ are discontinuous, it is in general preferable to use H(curl)-conforming finite element subspaces based on edge elements (see Chapter 15) instead of using H 1 -conforming finite element subspaces; see §45.4.

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Section: Putting Everything Together and Recalling The Identity From Stepmentioning

confidence: 99%

Finite Elements III

Ern

Guermond

2021

Texts in Applied Mathematics

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“…Convergence of SNI is well-known; see, e.g., [12,Theorem 2.5] and [34]. The SNI was also used as an inner iteration for the inexact Uzawa method [32] and the Krylov subspace method [29].…”

Section: First-order Problemsmentioning

confidence: 99%

A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

Li¹,

Wang²,

Wu³

et al. 2021

Preprint

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In this paper, we propose a direct parallel-in-time (PinT) algorithm for timedependent problems with first-or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix V grows as n is increased, where n is the number of time levels. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for computing V and V −1 , by which we prove that Cond 2 (V ) = O(n 2 ). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

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“…Preconditioner at the 'trial side'. The preconditioner presented in this section is inspired by constructions of preconditioners in [And16,NS19] for parabolic problems discretized on a tensor product of temporal and spatial spaces.…”

Section: Wavelets In Timementioning

confidence: 99%

A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations

Stevenson,

van Venetië,

Westerdiep

2021

Preprint

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In this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our theoretical findings.

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Time-Parallel Iterative Solvers for Parabolic Evolution Equations

Cited by 19 publications

References 42 publications

Finite Elements III

Finite Elements III

A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations

A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations

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