On the Neumann Laplacian in nonuniformly collapsing strips

2019 ◽  
Vol 22 (04) ◽  
pp. 1950021
Author(s):  
César R. de Oliveira ◽  
Alessandra A. Verri
Keyword(s):  
Boundary Conditions ◽  
Smooth Function ◽  
Robin Boundary Conditions ◽  
Effective Operator ◽  
Resolvent Convergence ◽  
Norm Resolvent Convergence ◽  
Neumann Laplacian ◽  
Robin Boundary ◽  
Effective Operators

Consider the Neumann Laplacian in the region below the graph of [Formula: see text], for a positive smooth function [Formula: see text] with both [Formula: see text] and [Formula: see text] bounded. As [Formula: see text] such region collapses to [Formula: see text] and an effective operator is found, which has Robin boundary conditions at [Formula: see text]. Then, we recover (under suitable assumptions in the case of unbounded [Formula: see text]) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for [Formula: see text] diverging less than exponentially.

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

scholarly journals Logarithmic minimal models with Robin boundary conditions

2016 ◽  
Vol 2016 (6) ◽  
pp. 063104 ◽  
Author(s):  
Jean-Emile Bourgine ◽  
Paul A Pearce ◽  
Elena Tartaglia
Keyword(s):  
Boundary Conditions ◽  
Robin Boundary Conditions ◽  
Minimal Models ◽  
Robin Boundary
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