Finite Elements III

Ern, Alexandre; Guermond, Jean‐Luc

doi:10.1007/978-3-030-57348-5

Cited by 23 publications

(8 citation statements)

References 204 publications

(304 reference statements)

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“…We first introduce a time reconstruction operator to rewrite the bilinear form b τ . This operator is classical in the context of dG methods in time; see, e.g., [19,Section 69.2.3] or [25, Section 2.3] and the references therein. Then, we study the stablity properties of A τ h in a suitable residual norm, we introduce interpolation operators in space and in time, and we bound the consistency error.…”

Section: Discussionmentioning

confidence: 99%

The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method

Burman,

Delay,

Ern

2023

SIAM J. Numer. Anal.

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We are interested in solving the unique continuation problem for the heat equation, i.e., we want to reconstruct the solution of the heat equation in a target space-time subdomain given its (noised) value in a subset of the computational domain. Both initial and boundary data can be unknown. We discretize this problem using a spacetime discontinuous Galerkin method (including hybrid variables in space) and look for the solution that minimizes a discrete Lagrangian. We establish discrete inf-sup stability and bound the consistency error, leading to a priori estimates on the residual. Owing to the ill-posed nature of the problem, an additional estimate on the residual dual norm is needed to prove the convergence of the discrete solution to the exact solution in the energy norm in the target space-time subdomain. This is achieved by combining the above results with a conditional stability estimate at the continuous level. The rate of convergence depends on the conditional stability, the approximation order in space and in time, and the size of the perturbations in data. Quite importantly, the weight of the regularization term depends on the time step and the mesh size, and we show how to choose it to preserve the best possible decay rates on the error. Finally, we run numerical simulations to assess the performance of the method in practice.

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Section: Discussionmentioning

confidence: 99%

The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method

Burman,

Delay,

Ern

2023

SIAM J. Numer. Anal.

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“…Proof This result is a consequence of the stability of Theorem 1, the consistency and (2.12). It is standard material (see [22,Section 76.4]) however for completeness we include the short proof. Using standard approximation estimates there holds [9, Lemma 5.6]…”

Section: Also Recall the Following Inverse Inequalitymentioning

confidence: 99%

“…In such situations, if continuous finite element spaces are to be used, one must ressort to a stabilized method to avoid a reduction of accuracy due to spurious oscillations. There is a very wide literature on stabilized methods and for an overview of the topic see for example [22]. In the high order case, the Spectral Vanishing Velocity method has been a popular choice [31,30,32], but other methods have also been designed to work in the high order case, see the discussion in [16].…”

Section: Introductionmentioning

confidence: 99%

Weighted error estimates for transient transport problems discretized using continuous finite elements with interior penalty stabilization on the gradient jumps

Burman¹

2021

Preprint

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In this paper we consider the semi-discretization in space of a first order scalar transport equation. For the space discretization we use standard continuous finite elements. To obtain stability we add a penalty on the jump of the gradient over element faces. We recall some global error estimates for smooth and rough solutions and then prove a new local error estimate for the transient linear transport equation. In particular we show that in the stabilized method the effect of non-smooth features in the solution decay exponentially from the space time zone where the solution is rough so that smooth features will be transported unperturbed. Locally the L 2 -norm error converges with the expected order O(h k+ 1 2 ). We then illustrate the results numerically. In particular we show the good local accuracy in the smooth zone of the stabilized method and that the standard Galerkin fails to approximate a solution that is smooth at the final time if discontinuities have been present in the solution at some time during the evolution.

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“…In all the numerical experiments reported in this work, the initial pressure is indeed known. If this were not the case, a possibility proposed in [45] (see also [36,Remarks 74.4 and 75.7]) is to recover the initial pressure by solving the following steady problem:…”

Section: Ac Schemementioning

confidence: 99%

“…The initialization of the iterative procedure is done with an approximation of the solution, for instance u n−1 h (as for the first-order scheme), or by using the second-order extrapolation formula (2 u n−1 h − u n−2 h ). Finally, for the BDF2 time discretization, it is possible to derive by means of classical algebraic manipulations (detailed, e.g., in [36,Lemma 68.1] for the heat equation) a time-discrete kinetic energy balance with dissipation, in the same spirit as in (26).…”

Section: Monolithic Schemementioning

confidence: 99%

Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

Milani

Bonelle²,

Ern

2021

Computational Methods in Applied Mathematics

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We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

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Finite Elements III

Cited by 23 publications

References 204 publications

The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method

The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method

Weighted error estimates for transient transport problems discretized using continuous finite elements with interior penalty stabilization on the gradient jumps

Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

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