Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

Nochetto, Ricardo H.; Otárola, Enrique; Salgado, Abner J.

doi:10.1007/s00211-015-0709-6

Cited by 60 publications

(116 citation statements)

References 70 publications

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“…It is known that the numerical approximation of functions with a strong directional-dependent behavior needs anisotropic elements in order to recover quasi-optimal error estimates [7,21]. In our setting, anisotropic elements of tensor product structure are essential.…”

Section: Finite Element Methodsmentioning

confidence: 99%

A PDE Approach to Space-Time Fractional Parabolic Problems

Nochetto¹,

Otárola²,

Salgado³

2016

SIAM J. Numer. Anal.

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127

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Abstract. We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully-discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.

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Section: Finite Element Methodsmentioning

confidence: 99%

A PDE Approach to Space-Time Fractional Parabolic Problems

Nochetto¹,

Otárola²,

Salgado³

2016

SIAM J. Numer. Anal.

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118

127

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“…It is still unclear if the same techniques may be used for the treatment of jumps in the solution itself, as in these cases the regularity gain that could be achieved by weighted Sobolev spaces alone may not be sufficient, and we are currently exploring alternative approximation frameworks, following the lines of [39].…”

Section: Discussionmentioning

confidence: 99%

Error estimates in weighted Sobolev norms for finite element immersed interface methods

Heltai

Rotundo

2019

Computers & Mathematics with Applications

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When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation.A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods.In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.

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“…This condition allows for anisotropy in the extended variable y [11,16,17], which is needed to compensate the rather singular behavior of , solution to (3.3). We refer the reader to [11] for details.…”

Section: A Fully Discrete Scheme For the Fractional Optimal Control Pmentioning

confidence: 99%

“…We shall refer to y as the extended variable and to the dimension n + 1, in R n+1 + , the extended dimension of problem (1.7). The main advantage of the scheme proposed in [11] is that it involves the resolution of the local problem (1.7) and thus its implementation uses basic ingredients of finite element analysis; its analysis, however, involves asymptotic estimates of Bessel functions [13], to derive regularity estimates in weighted Sobolev spaces, elements of harmonic analysis [14,15], and an anisotropic polynomial interpolation theory in weighted Sobolev spaces [16,17]. The use of the extension problem (1.7) for the discretization of the spectral fractional Laplacian was first used in [11].…”

Section: Introductionmentioning

confidence: 99%

“…The use of the extension problem (1.7) for the discretization of the spectral fractional Laplacian was first used in [11]. The main advantage of the scheme proposed in [11] is that it involves the resolution of the local problem (1.7) and thus its implementation uses basic ingredients of finite element analysis; its analysis, however, involves asymptotic estimates of Bessel functions [13], to derive regularity estimates in weighted Sobolev spaces, elements of harmonic analysis [14,15], and an anisotropic polynomial interpolation theory in weighted Sobolev spaces [16,17]. Such an interpolation theory allows for tensor product elements that exhibit an anisotropic feature in the extended dimension, that is in turn needed to compensate the singular behavior of the solution in the extended variable y ( [11], Theorem 2.7, [10], Theorem 4.7]).…”

Section: Introductionmentioning

confidence: 99%

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An adaptive finite element method for the sparse optimal control of fractional diffusion

Otárola

2019

Numerical Methods Partial

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We propose and analyze an a posteriori error estimator for a partial differential equation (PDE)-constrained optimization problem involving a nondifferentiable cost functional, fractional diffusion, and control-constraints. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly PDE and propose an equivalent optimal control problem with a local state equation. For such an equivalent problem, we design an a posteriori error estimator which can be defined as the sum of four contributions: two contributions related to the approximation of the state and adjoint equations and two contributions that account for the discretization of the control variable and its associated subgradient. The contributions related to the discretization of the state and adjoint equations rely on anisotropic error estimators in weighted Sobolev spaces. We prove that the proposed a posteriori error estimator is locally efficient and, under suitable assumptions, reliable. We design an adaptive scheme that yields, for the examples that we perform, optimal experimental rates of convergence. KEYWORDSa posteriori error analysis, adaptive loop, anisotropic estimates, nonlocal operators, nonsmooth objectives, PDE-constrained optimization, sparse controls, spectral fractional Laplacian Enrique Otárola is partially supported by CONICYT through FONDECYT project 11180193.Numer Methods Partial Differential Eq. 2020;36:302-328. wileyonlinelibrary.com/journal/num

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Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

Cited by 60 publications

References 70 publications

A PDE Approach to Space-Time Fractional Parabolic Problems

A PDE Approach to Space-Time Fractional Parabolic Problems

Error estimates in weighted Sobolev norms for finite element immersed interface methods

An adaptive finite element method for the sparse optimal control of fractional diffusion

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