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See detailA posteriori error estimation for finite element approximations of fractional Laplacian problems and applications to poro–elasticity
Bulle, Raphaël UL

Doctoral thesis (2022)

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional ... [more ▼]

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posteriori error estimation in finite element methods and of adaptive refine- ment methods. We emphasize the state–of–the–art of the Bank–Weiser a posteriori error estimation method and of the adaptive refinement methods convergence results. Then, we move to fractional partial differential equations. We give some of the most common discretization methods of fractional Laplacian operator based equations. We review some results of a priori error estimation for the finite element discretization of these equations and give the state–of–the–art of a posteriori error estimation. Finally, we review the literature on the use of the Caputo’s fractional derivative in applications, focusing on anomalous diffusion and poro-elasticity applications. The rest of the manuscript is organized as follow. Chapter 1 is concerned with a proof of the reliability of the Bank–Weiser estimator for three–dimensional problems, extending a result from the literature. In Chapter 2 we present a numerical study of the Bank–Weiser estimator, provide a novel implementation of the estimator in the FEniCS finite element software and apply it to a variety of elliptic equations as well as goal-oriented error estimation. In Chapter 3 we derive a novel a posteriori estimator for the L2 error induced by the finite element discretization of fractional Laplacian operator based equations. In Chapter 4 we present new theoretical results on the convergence of a rational approximation method with consequences on the approximation of fractional norms as well as a priori error estimation results for the finite element discretization of fractional equations. Finally, in Chapter 5 we provide an application of fractional calculus to the study of the human meniscus via poro-elasticity equations. [less ▲]

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See detailAn a posteriori error estimator for the spectral fractional power of the Laplacian
Bulle, Raphaël UL; Barrera, Olga; Bordas, Stéphane UL et al

E-print/Working paper (2022)

We develop a novel a posteriori error estimator for the L2 error committed by the finite ele- ment discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes ... [more ▼]

We develop a novel a posteriori error estimator for the L2 error committed by the finite ele- ment discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi–discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non–fractional parametric problems. The estimator involves applying the implicit Bank–Weiser error estimation strategy to each parametric non–fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy. [less ▲]

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See detailHierarchical a posteriori error estimation of Bank-Weiser type in the FEniCS Project
Bulle, Raphaël UL; Hale, Jack UL; Lozinski, Alexei et al

E-print/Working paper (2021)

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of ... [more ▼]

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material. [less ▲]

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See detailRemoving the saturation assumption in Bank-Weiser error estimator analysis in dimension three
Bulle, Raphaël UL; Chouly, Franz; Hale, Jack UL et al

in Applied Mathematics Letters (2020), 107

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes ... [more ▼]

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional test case. [less ▲]

Detailed reference viewed: 164 (41 UL)