Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

2014 ◽  
Vol 6 (5) ◽  
pp. 604-614 ◽  
Author(s):  
Mei-Ling Sun ◽  
Shan Jiang
Keyword(s):  
Numerical Experiments ◽  
Computational Cost ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Basis Functions ◽  
Order Convergence ◽  
Numerical Accuracy ◽  
First Order ◽  
Graded Meshes ◽  
Second Order Convergence

AbstractWe apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in theL2norm and first order convergence in the energy norm on graded meshes, which is independent of ɛ. In contrast with the conventional methods, our method is much more accurate and effective.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

Analysis of three-layer composite plates with a new higher-order layerwise formulation

2014 ◽  
Vol 21 (3) ◽  
pp. 401-404
Author(s):  
Dalal A. Maturi ◽  
Antonio J.M. Ferreira ◽  
Ashraf M. Zenkour ◽  
Daoud S. Mashat
Keyword(s):  
Composite Plates ◽  
Order Theory ◽  
Higher Order ◽  
Basis Functions ◽  
Numerical Accuracy ◽  
First Order ◽  
The Core ◽  
First Order Theory ◽  
Layer Composite ◽  
Vibration Problems

AbstractIn this paper, we combine a new higher-order layerwise formulation and collocation with radial basis functions for predicting the static deformations and free vibration behavior of three-layer composite plates. The skins are modeled via a first-order theory, while the core is modeled by a cubic expansion with the thickness coordinate. Through numerical experiments, the numerical accuracy of this strong-form technique for static and vibration problems is discussed.

scholarly journals COMPARISON OF ADAPTIVE MESHES FOR A SINGULARLY PERTURBED REACTION–DIFFUSION PROBLEM

2012 ◽  
Vol 17 (5) ◽  
pp. 732-748 ◽  
Author(s):  
Andrej Bugajev ◽  
Raimondas Čiegis
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Adaptive Mesh ◽  
Diffusion Problem ◽  
Adaptive Meshes ◽  
Finite Element Scheme ◽  
Differential Problem ◽  
Galerkin Finite Element ◽  
Error Estimators ◽  
Shishkin Meshes

We consider a singular second-order boundary value problem. The differential problem is approximated by the Galerkin finite element scheme. The main goal is to compare the well known apriori Bakhvalov and Shishkin meshes with the adaptive mesh based on the aposteriori dual error estimators. Results of numerical experiments are presented.

scholarly journals A Collocation Method for Numerical Solution of Hyperbolic Telegraph Equation with Neumann Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. C. Mittal ◽  
Rachna Bhatia
Keyword(s):  
Boundary Conditions ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Telegraph Equation ◽  
Neumann Boundary Conditions ◽  
Basis Functions ◽  
One Dimensional ◽  
First Order ◽  
Spline Basis ◽  
Good Agreement

We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.

Singularities in p-Type Finite Element Computations: Numerical Experiments on the Locality of Pollution Effects

1991 ◽  
Vol 58 (2) ◽  
pp. 586-588 ◽  
Author(s):  
G. Becker ◽  
P. Carnevali ◽  
B. Chayapathy ◽  
W. Imaino ◽  
R. B. Morris ◽  
...  
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Basis Functions ◽  
Effective Wavelength ◽  
Pollution Effects ◽  
Element Size ◽  
Graded Meshes ◽  
P Type

Using a recently developed code for three-dimensional p-type finite element computations in elasticity, we have studied, with a series of numerical experiments, the behavior of the solution in the presence of singularities. Worst-cast theoretical estimates predict a global pollution of the solution, unless optimally graded meshes are used. On the other hand, we observe that except in an immediate neighborhood of a singularity, and even using uniform meshes, the quality of the solution is not degraded to any practically relevant extent. This has important practical consequences because it allows modelers to introduce artifical singularities for the purpose of simplifying models. The size of the area around a singularity where the solution is appreciably degraded can be estimated in terms of the minimum effective wavelength of the basis functions used; the latter, in turn, can be related to the element size and the polynomial order used.

scholarly journals A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Musa Çakır ◽  
Gabil M. Amiraliyev
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Convection Diffusion ◽  
First Order ◽  
Type Boundary

The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameterε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.

On Finite Volume Discretization of the Three-dimensional Biot Poroelasticity System in Multilayer Domains

2006 ◽  
Vol 6 (3) ◽  
pp. 306-325 ◽  
Author(s):  
A. Naumovich
Keyword(s):  
Physical Properties ◽  
Finite Volume ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Second Order ◽  
Maximum Norm ◽  
Order Convergence ◽  
Staggered Grids ◽  
Finite Volume Discretization ◽  
Second Order Convergence

AbstractIn this paper we propose a finite volume discretization for the threedimensional Biot poroelasticity system in multilayer domains. For stability reasons, staggered grids are used. The discretization takes into account discontinuity of the coefficients across the interfaces between layers with different physical properties. Numerical experiments based on the proposed discretization showed second order convergence in the maximum norm for the primary and flux unknowns of the system. An application example is presented as well.

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