Optimal Convergence for the Implicit Space‐Time Discretization of Parabolic Systems with <i>p</i>‐Structure

Diening, Lars; Ebmeyer, Carsten; ring, Michael Ru; caron, Zdene; caron, ic; ka,

doi:10.1137/05064120x

Cited by 61 publications

(59 citation statements)

References 13 publications

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“…In this case, the bound (22) (with possibly different constant C) can be directly derived from (20). Hence, the lower bound (22) for a ξ K (v; v) holds for any v ∈ W (K δ ) and, in particular, a ξ K is coercive on W (K δ ) for both periodic and Dirichlet coupling.…”

Section: Existence and Uniqueness Of The Numerical Solutionmentioning

confidence: 97%

“…Note however that for p = 2, higher regularity of u 0 is not realistic even for smooth data, e.g., see [11,20]. …”

Section: Theorem 42 Assume That Amentioning

confidence: 99%

“…Let us close the introduction by putting our results in contrast to existing FE approximation results of single scale parabolic monotone problems. In the L p (W 1,p ) setting, optimal explicit convergence rates in terms of the discretization parameters have been derived for maps with a p-structure 3 , e.g., the parabolic p-Laplacian, using quasi-norms in space, see [11,20]. Note however that under the assumptions (A 1−2 ) the maps A ε have p-structure if and only if α = 1 and β = 2.…”

mentioning

confidence: 99%

“…As we assume 0 < α ≤ min{p − 1, 1} and max{2, p} ≤ β < ∞ (the most general assumptions on the oscillatory maps allowing for homogenization, see [15,18,42]), we have in addition that p = 2 if we want both a p-structure and a valid homogenization setting. For this set of parameters, the quasi-norm (in space) from [11,20] collapses to the standard H 1 (Ω) norm. For all other values of p, homogenization theory seems not to exist for maps A ε with p-structure and thus studying numerical homogenization methods makes no sense.…”

mentioning

confidence: 99%

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Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

Abdulle

Huber

2016

Numer. Methods Partial Differential Eq.

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We propose a multiscale method based on a finite element heterogeneous multiscale method (in space) and the implicit Euler integrator (in time) to solve nonlinear monotone parabolic problems with multiple scales due to spatial heterogeneities varying rapidly at a microscopic scale. The multiscale method approximates the homogenized solution at computational cost independent of the small scale by performing numerical upscaling (coupling of macro and micro finite element methods). Taking into account the error due to time discretization as well as macro and micro spatial discretizations, the convergence of the method is proved in the general L p (W 1,p ) setting. For p = 2, optimal convergence rates in the L 2 (H 1 ) and C 0 (L 2 ) norm are derived. Numerical experiments illustrate the theoretical error estimates and the applicability of the multiscale method to practical problems.

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Section: Existence and Uniqueness Of The Numerical Solutionmentioning

confidence: 97%

“…Note however that for p = 2, higher regularity of u 0 is not realistic even for smooth data, e.g., see [11,20]. …”

Section: Theorem 42 Assume That Amentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

Abdulle

Huber

2016

Numer. Methods Partial Differential Eq.

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“…Such degenerated parabolic problems are particularly challenging numerically and for the analysis, due to the poor regularity of the exact solutions, see e.g. [10,14]. However, the homogenization results of [36,38] cited in Section 2 (for monotone operators on H 1 (Ω)) hold as well for monotone operators on W 1,p (Ω) for p ≥ 2, e.g., for operators with nonlinearities similar to the p-Laplacian.…”

Section: Case Of a Degenerated Problemmentioning

confidence: 99%

Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems

Abdulle¹,

Huber²,

Vilmart³

2015

Multiscale Model. Simul.

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We introduce and analyze an efficient numerical homogenization method for a class of nonlinear parabolic problems of monotone type in highly oscillatory media. The new scheme avoids costly Newton iterations and is linear at both the macroscopic and the microscopic scales. It can be interpreted as a linearized version of a standard nonlinear homogenization method. We prove the stability of the method and derive optimal a priori error estimates which are fully discrete in time and space. Numerical experiments confirm the error bounds and illustrate the efficiency of the method for various nonlinear problems.Keywords: monotone parabolic multiscale problem, linearized scheme, numerical homogenization method, fully discrete a priori error estimates.

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On uniqueness and time regularity of flows of power-law like non-Newtonian fluids

Bulíček

Ettwein

Kaplický

et al. 2010

Math. Meth. Appl. Sci.

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Large class of non-Newtonian fluids can be characterized by index p, which gives the growth of the constitutively determined part of the Cauchy stress tensor. In this paper, the uniqueness and the time regularity of flows of these fluids in an open bounded three-dimensional domain is established for subcritical ps, i.e. for p>11 / 5. Our method works for 'all' physically relevant boundary conditions, the Cauchy stress need not be potential and it may depend explicitly on spatial and time variable. As a simple consequence of time regularity, pressure can be introduced as an integrable function even for Dirichlet boundary conditions. Moreover, these results allow us to define a dynamical system corresponding to the problem and to establish the existence of an exponential attractor. Copyright © 2010 John Wiley & Sons, Ltd.Keywords: non-Newtonian fluids; time regularity; uniqueness; exponential attractor The problem and the main resultsLet us consider an incompressible, homogeneous fluid, which is characterized by Newton's rheological lawwhere S is the constitutively determined part of the Cauchy stress tensor, Dv the symmetric velocity gradient. One should assume that the viscosity >0 is a constant, we obtain the so-called Navier-Stokes equations (NSE). There is no doubt that, at least in certain regimes, this is a reasonable model, and the NSEs belong to the most studied systems of mathematical physics.On the other hand, there is both theoretical and experimental evidence suggesting that the viscous forces can be effective function of other physical quantities; the pressure, density, electric field and last, but not the least, the shear rate |Dv|. We refer to the survey article [1] for the detailed discussion of both the physical background and the state of art of the available mathematical results.In this paper, we focus on qualitative properties of flows of such fluids, more precisely, on the uniqueness, regularity and also large-time behavior of weak solutions to the following system of equations:that are supposed to be satisfied in a space-time cylinder Q := I× , where I ⊂ R denotes an open bounded time interval and ⊂ R 3 is an open bounded set. The first equation in (1) represents the balance of linear momentum driven by the given external body forces f , while the second one can be considered as the balance of mass for homogeneous incompressible fluid. The internal properties of the fluid are given by the constitutively determined part of the Cauchy stress tensor S. The classical examples of the fluids that can be treated are described by the so-called power-law model, where S is given by the relation S = S(Dv):= 0 |Dv| p−2 Dv, and its non-degenerate variant, the so-called Ladyzhenskaya model, where S takes the form S := 0 (1+|Dv| 2 ) (p−2)/2 Dv.From the mathematical point of view, for power-law like fluid one formally obtains the L p -estimates on ∇v, which (for p>2) are a certain remedy to the well-known lack of (the information about) the regularity of solutions of the original Navier-Stokes problem,

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Optimal Convergence for the Implicit Space‐Time Discretization of Parabolic Systems with p‐Structure

Cited by 61 publications

References 13 publications

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems

On uniqueness and time regularity of flows of power-law like non-Newtonian fluids

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