Displaying similar documents to “Local Discontinuous Galerkin methods for fractional diffusion equations”

Robust operator estimates and the application to substructuring methods for first-order systems

Christian Wieners, Barbara Wohlmuth (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz , 49 (2011) 1788–1809; Zitelli , 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space. Here, we show that the abstract...

On shape optimization problems involving the fractional laplacian

Anne-Laure Dalibard, David Gérard-Varet (2013)

ESAIM: Control, Optimisation and Calculus of Variations

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Our concern is the computation of optimal shapes in problems involving (−). We focus on the energy (Ω) associated to the solution of the basic Dirichlet problem ( − )  = 1 in Ω,  = 0 in Ω. We show that regular minimizers Ω of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the...

Optimal convergence rates of mortar finite element methods for second-order elliptic problems

Faker Ben Belgacem, Padmanabhan Seshaiyer, Manil Suri (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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We present an improved, near-optimal error estimate for a non-conforming finite element method, called the mortar method (M0). We also present a new mortaring technique, called the mortar method (MP), and derive , and error estimates for it, in the presence of quasiuniform and non-quasiuniform meshes. Our theoretical results, augmented by the computational evidence we present, show that like (M0), (MP) is also a viable mortaring technique for the method.

α-time fractional Brownian motion: PDE connections and local times

Erkan Nane, Dongsheng Wu, Yimin Xiao (2012)

ESAIM: Probability and Statistics

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For 0 <  ≤ 2 and 0 <  < 1, an -time fractional Brownian motion is an iterated process  =  {() = (()) ≥ 0}  obtained by taking a fractional Brownian motion  {() ∈ ℝ} with Hurst index 0 <  < 1 and replacing the time parameter with a strictly -stable Lévy process {() ≥ 0} in ℝ independent of {() ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when ...

α-time fractional brownian motion: PDE connections and local times

Erkan Nane, Dongsheng Wu, Yimin Xiao (2012)

ESAIM: Probability and Statistics

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For 0 &lt;  ≤ 2 and 0 &lt;  &lt; 1, an -time fractional Brownian motion is an iterated process  =  {() = (()) ≥ 0}  obtained by taking a fractional Brownian motion  {() ∈ ℝ} with Hurst index 0 &lt;  &lt; 1 and replacing the time parameter with a strictly -stable Lévy process {() ≥ 0} in ℝ independent of {() ∈ R}. It is shown that such processes have natural connections to partial differential equations and, when is a stable subordinator, can arise as scaling limit...

Optimal convergence of a discontinuous-Galerkin-based immersed boundary method

Adrian J. Lew, Matteo Negri (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

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We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional...

Optimal convergence of a discontinuous-Galerkin-based immersed boundary method

Adrian J. Lew, Matteo Negri (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

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We prove the optimal convergence of a discontinuous-Galerkin-based immersed boundary method introduced earlier [Lew and Buscaglia, (2008) 427–454]. By switching to a discontinuous Galerkin discretization near the boundary, this method overcomes the suboptimal convergence rate that may arise in immersed boundary methods when strongly imposing essential boundary conditions. We consider a model Poisson's problem with homogeneous boundary conditions over two-dimensional...

Error Control and Andaptivity for a Phase Relaxation Model

Zhiming Chen, Ricardo H. Nochetto, Alfred Schmidt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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The phase relaxation model is a diffuse interface model with small parameter which consists of a parabolic PDE for temperature and an ODE with double obstacles for phase variable . To decouple the system a semi-explicit Euler method with variable step-size is used for time discretization, which requires the stability constraint . Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter are further employed for space discretization. error estimates...