scholarly journals Convergence rates for the classical, thin and fractional elliptic obstacle problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140449 ◽  
Author(s):  
Ricardo H. Nochetto ◽  
Enrique Otárola ◽  
Abner J. Salgado
Keyword(s):  
Obstacle Problem ◽  
Fractional Laplacian ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Obstacle Problems ◽  
Element Approximation ◽  
Optimal Error ◽  
Optimal Convergence Rates ◽  
Regularity Results ◽  
Thin Obstacle

We review the finite-element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results, we present an optimal error analysis for the thin obstacle problem. Finally, we discuss the localization of the obstacle problem for the fractional Laplacian and prove quasi-optimal convergence rates.

scholarly journals Finite element approximation of an obstacle problem for a class of integro–differential operators

2020 ◽  
Vol 54 (1) ◽  
pp. 229-253 ◽  
Author(s):  
Andrea Bonito ◽  
Wenyu Lei ◽  
Abner J. Salgado
Keyword(s):  
Finite Element ◽  
Obstacle Problem ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Finite Element Scheme ◽  
Order Elliptic Operator ◽  
Nonlocal Part ◽  
Differential Part

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

scholarly journals Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters

2014 ◽  
Vol 14 (4) ◽  
pp. 509-535 ◽  
Author(s):  
Dietmar Gallistl
Keyword(s):  
Finite Element ◽  
Convergence Rates ◽  
Nonlinear Approximation ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
Eigenvalues Of The Laplacian ◽  
Optimal Convergence Rates ◽  
Approximation Classes ◽  
Adjoint Operators

AbstractThis paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.

scholarly journals ON GLOBAL SPATIAL REGULARITY AND CONVERGENCE RATES FOR TIME-DEPENDENT ELASTO-PLASTICITY

2010 ◽  
Vol 20 (10) ◽  
pp. 1823-1858 ◽  
Author(s):  
DOROTHEE KNEES
Keyword(s):  
Convergence Rates ◽  
Regularity Result ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Internal Variables ◽  
Element Approximation ◽  
Elasto Plasticity ◽  
Spatial Regularity ◽  
Regularity Results ◽  
Tangential Regularity

We study the global spatial regularity of solutions of generalized elasto-plastic models with linear hardening on smooth domains. Under natural smoothness assumptions on the data and the boundary we obtain u ∈ L∞((0, T); H3/2-δ(Ω)) for the displacements and z ∈ L∞((0, T); H1/2-δ(Ω)) for the internal variables. The proof relies on a reflection argument which gives the regularity result in directions normal to the boundary on the basis of tangential regularity results. Based on the regularity results we derive convergence rates for a finite element approximation of the models.

scholarly journals Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

scholarly journals Regularity results of the thin obstacle problem for the p(x)-Laplacian

2019 ◽  
Vol 276 (2) ◽  
pp. 496-519 ◽  
Author(s):  
Sun-Sig Byun ◽  
Ki-Ahm Lee ◽  
Jehan Oh ◽  
Jinwan Park
Keyword(s):  
Obstacle Problem ◽  
Regularity Results ◽  
Thin Obstacle

Robust Approximation of Singularly Perturbed Delay Differential Equations by the hp Finite Element Method

2013 ◽  
Vol 13 (1) ◽  
pp. 21-37 ◽  
Author(s):  
Serge Nicaise ◽  
Christos Xenophontos
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Convergence Rates ◽  
Finite Element Approximation ◽  
Interface Layer ◽  
Singularly Perturbed ◽  
Element Approximation ◽  
Smooth Part ◽  
Delay Differential ◽  
Element Method

Abstract. We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

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