Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems

Ern, Alexandre; Smears, Iain; Vohralı́k, Martin

doi:10.1137/16m1097626

Cited by 37 publications

(58 citation statements)

References 38 publications

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“…The continuous Petrov-Galerkin method in time with linear trial and constant test functions (denoted cGP (1)) is equivalent to the Crank-Nicholson method. Recently, higher order dG(k)-and cGP(k)-methods have been developed and analyzed for parabolic and hyperbolic problems [25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Variational formulations for explicit Runge-Kutta Methods

Muñoz‐Matute

Pardo

Calo

et al. 2019

Finite Elements in Analysis and Design

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Variational space-time formulations for Partial Differential Equations have been of great interest in the last decades. While it is known that implicit time marching schemes have variational structure, the Galerkin formulation of explicit methods in time remains elusive. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous Petrov-Galerkin methods both in space and time. We build trial and test spaces for the linear diffusion equation that lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to design explicit time-domain (goal-oriented) adaptive algorithms.

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Section: Introductionmentioning

confidence: 99%

Variational formulations for explicit Runge-Kutta Methods

Muñoz‐Matute

Pardo

Calo

et al. 2019

Finite Elements in Analysis and Design

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“…Remark 2.4 (Heat equation). Sharp estimates of the inf-sup constant β for the heat equation (with µ(t) ≡ 1 on I so that α = M = 1 and κ = 0) can be found in [36,10] using the above norms.…”

Section: Moreover Using Again the Coercivity Of A(t) And The Boundedmentioning

confidence: 99%

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

et al. 2019

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We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert-Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov-Galerkin setting to evaluate the dual residual norm.

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“…A := T 0 ∇v 2 Ω dt, with · Ω denoting the L 2 -norm over Ω (for proof, see for instance [9]). For each trial function u ∈ S, there is an optimal choice of test function v ∈ L 2 (0, T ; H 1 0 (Ω)) that achieves the supremum in (1.1); in the discrete setting, this mapping between trial and optimal test functions leads to a left-preconditioner that symmetrizes the system in a stable way.…”

Section: Introductionmentioning

confidence: 99%

“…We note that the inf-sup theory for parabolic problems has previously found application in other contexts, such as a priori error analysis [35], a posteriori analysis [9], and reduced-basis methods [36]. We also refer the reader to the textbooks [8,33] for an introduction to the inf-sup theorem for general linear equations in Banach spaces, and its application to parabolic problems.…”

Section: Introductionmentioning

confidence: 99%

Time-Parallel Iterative Solvers for Parabolic Evolution Equations

Neumüller¹,

Smears²

2019

SIAM J. Sci. Comput.

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We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties.

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Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems

Cited by 37 publications

References 38 publications

Variational formulations for explicit Runge-Kutta Methods

Variational formulations for explicit Runge-Kutta Methods

Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

Time-Parallel Iterative Solvers for Parabolic Evolution Equations

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