On the Sobolev and $L^p$-Stability of the $L^2$-Projection

Diening, Lars; Storn, Johannes; Tscherpel, Tabea

doi:10.1137/20m1358013

Cited by 13 publications

(12 citation statements)

References 24 publications

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“…It is known that this stability holds also true for (shape regular) locally refined partitions when they are sufficiently mildly graded. In [GHS16], it is shown that in two space dimensions the meshes generated by newest vertex bisection satisfy this requirement, see also [DST20] for extensions.…”

Section: Brézis-ekeland-nayroles (Ben) Formulationmentioning

confidence: 99%

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

Stevenson¹,

Westerdiep²

2021

Preprint

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We consider a minimal residual discretization of a simultaneous spacetime variational formulation of parabolic evolution equations. Under the usual 'LBB' stability condition on pairs of trial-and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm which are independent of this spatial differential operator.

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Section: Brézis-ekeland-nayroles (Ben) Formulationmentioning

confidence: 99%

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

Stevenson¹,

Westerdiep²

2021

Preprint

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“…In fact, for special meshes one can show also the W 1,p stability. The improved stability for the L 2 -projection has a long history, see Douglas, Jr., Dupont, and Wahlbin [14], Bramble and Xu [9], and the review in the recent work of Diening, Storn, and Tscherpel [13]. A different approach in Hilbert fractional spaces is used in [16], but we do not known whether this applies to the estimates we are willing to use.…”

Section: Moreover This Term Can Be Estimated As Followsmentioning

confidence: 99%

On the convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations

Berselli¹,

Spirito²

2022

Preprint

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We prove the convergence of certain second-order numerical methods to weak solutions of the Navier-Stokes equations satisfying in addition the local energy inequality, and therefore suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, we treat the space-periodic case in three space-dimensions and we consider a full discretization in which the the classical Crank-Nicolson method (θ-method with θ = 1/2) is used to discretize the time variable, while in the space variables we consider finite elements. The convective term is discretized in several implicit, semi-implicit, and explicit ways. In particular, we focus on proving (possibly conditional) convergence of the discrete solutions towards weak solutions (satisfying a precise local energy balance), without extra regularity assumptions on the limit problem. We do not prove orders of convergence, but our analysis identifies some numerical schemes providing also alternate proofs of existence of "physically relevant" solutions in three space dimensions.

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“…n k i=1 . Algorithm 6 details a (nonrecursive) implementation of a single multiplicative V-cycle for the multilevel decomposition (24) with Gauss-Seidel smoothing. We assume the availability of an efficient coarse-grid solver; for us, a direct solve sufficed.…”

Section: Operations In Spacementioning

confidence: 99%

“…11 We are not aware of a similar result for

d>2

; this is one part of the reason for the restriction to

d=2

; the other is that for

d>2

, it has not been shown that triangulations generated by NVB are sufficiently mildly graded to ensure

{H}_0^1\left(\Omega \right)

‐stability of the

{L}_2\left(\Omega \right)

‐orthogonal projection, which is vital for the proof of Theorem 6 below; see Reference 24 for an in‐depth analysis.…”

mentioning

confidence: 99%

Efficient space‐time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space

Venetië

Westerdiep

2022

Numerical Linear Algebra App

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Considering the space-time adaptive method for parabolic evolution equations we introduced in Stevenson et al., this work discusses an implementation of the method in which every step is of linear complexity. Exploiting the tensor-product structure of the space-time cylinder, the method allows for a family of trial spaces given as spans of wavelets-in-time tensorized with finite element spaces-in-space. On spaces whose bases are indexed by double-trees, we derive an algorithm that applies the resulting bilinear forms in linear complexity.We provide extensive numerical experiments to demonstrate the linear runtime of the resulting adaptive loop.

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On the Sobolev and $L^p$-Stability of the $L^2$-Projection

Cited by 13 publications

References 24 publications

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

On the convergence of second-order in time numerical discretizations for the evolution Navier-Stokes equations

Efficient space‐time adaptivity for parabolic evolution equations using wavelets in time and finite elements in space

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