OPTIMAL INTERFACE ERROR ESTIMATES FOR A DISCRETE DOUBLE OBSTACLE APPROXIMATION TO THE PRESCRIBED CURVATURE PROBLEM

1999 ◽  
Vol 09 (06) ◽  
pp. 799-823 ◽  
Author(s):  
BARBARA CECON ◽  
MAURIZIO PAOLINI ◽  
MARIANGELA ROMEO
Keyword(s):  
Error Estimate ◽  
Piecewise Linear ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Element Discretization ◽  
The Maximum Principle ◽  
Formal Asymptotics ◽  
Prescribed Curvature ◽  
The Second Variation ◽  
Curvature Problem

In this paper we consider the so-called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend Ref. 10 with the introduction of the same nonregular potential used for the evolution problem in Ref. 9 and prove an optimal [Formula: see text] error estimate for nondegenerate minimizers (where ε represents the perturbation parameter). Following Ref. 10 the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error estimate for discrete minima using the Rannacher–Scott L∞-estimates and under appropriate restrictions on the mesh size ([Formula: see text] with σ>5/2).

scholarly journals Study of Post-Peak Strain Softening Mechanical Behaviour of Rock Material Based on Hoek–Brown Criterion

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Qibin Lin ◽  
Ping Cao ◽  
Peixin Wang
Keyword(s):  
Mechanical Behaviour ◽  
Strain Softening ◽  
Piecewise Linear ◽  
Peak Stress ◽  
Peak Hardness ◽  
Analytical Relationship ◽  
Peak Strain ◽  
Theoretical Derivation ◽  
The Maximum Principle ◽  
Laboratory Test Results

In order to build the post-peak strain softening model of rock, the evolution laws of rock parameters m,s were obtained by using the evolutionary mode of piecewise linear function regarding the maximum principle stress. Based on the nonlinear Hoek–Brown criterion, the analytical relationship of the rock strength parameters m,s, cohesion c, and friction angle φ has been developed by theoretical derivation. According to the analysis on the four different types of rock, it is found that, within the range from 0 to σ3min, the peak hardness of the rock becomes smaller as the confining pressure increases and the degree of rock fragmentation decreases as well. The post-peak stress-strain curves obtained from the developed softening model are in good agreement with the laboratory test results under different confining pressures. In conclusion, the analytical method is reasonable, and it can predict the post-peak mechanical behaviour of rock well, which provides a new thought for the rock-softening simulation.

CONVERGENCE OF A hp-STREAMLINE DIFFUSION SCHEME FOR VLASOV–FOKKER–PLANCK SYSTEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1159-1182 ◽  
Author(s):  
M. ASADZADEH ◽  
A. SOPASAKIS
Keyword(s):  
Finite Element ◽  
A Priori ◽  
Mesh Size ◽  
Stability Estimates ◽  
Spectral Order ◽  
Element Discretization ◽  
Diffusion Scheme ◽  
Streamline Diffusion ◽  
Stabilization Parameter ◽  
Fokker Planck

We analyze the hp-version of the streamline diffusion finite element method for the Vlasov–Fokker–Planck system. For this method we prove the stability estimates and derive sharp a priori error bounds in a stabilization parameter δ ~ min (h/p, h2/σ), with h denoting the mesh size of the finite element discretization in phase-space-time, p the spectral order of approximation, and σ the transport cross-section.

scholarly journals Error estimates for the unilateral buckling load of a plate involving the membrane efforts consistency error

2008 ◽  
pp. 1003-1038
Author(s):  
Mekki Ayadi
Keyword(s):  
Finite Elements ◽  
Error Estimate ◽  
Critical Load ◽  
Error Estimates ◽  
Thin Plate ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Buckling Load ◽  
Plate Model ◽  
Consistency Error

The paper deals with error estimates for the unilateral buckling critical load of a thin plate in presence of an obstacle. The error on the membrane efforts tensor is taken into account. First, using the Mindlin’s plate model together with a finite elements scheme of degree one, an error estimate, depending on the mesh size h, is established. In order to validate this theoretical error estimate, some numerical experiments are presented. Second, using the Kirchhoff-Love’s plate model, an abstract error estimate is achieved. Its drawback is that it contains a hard term to evaluate.

scholarly journals Numerical Solutions to Nonsmooth Dirichlet Problems Based on Lumped Mass Finite Element Discretization

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Haixiong Yu ◽  
Jinping Zeng
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Uniform Mesh ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Lumped Mass ◽  
Element Discretization ◽  
Maximum Angle

We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented.

AN ADAPTIVE FINITE ELEMENT METHOD WITH EFFICIENT MAXIMUM NORM ERROR CONTROL FOR ELLIPTIC PROBLEMS

1994 ◽  
Vol 04 (03) ◽  
pp. 313-329 ◽  
Author(s):  
KENNETH ERIKSSON
Keyword(s):  
Finite Element ◽  
Error Control ◽  
A Priori ◽  
Mesh Size ◽  
Maximum Norm ◽  
Adaptive Finite Element Method ◽  
Local Regularity ◽  
Adaptive Finite Element ◽  
Element Discretization ◽  
Elliptic Model

A posteriori and a priori error estimates are derived for a finite element discretization method applied to an elliptic model problem. The underlying partitions need not be quasi-uniform and can be highly graded; only a certain weak, local mesh regularity is assumed. The error is bounded in terms of the local mesh size and the local regularity of the solution and data. An adaptive algorithm is designed for automatic control of the discretization error in the maximum norm. The error control is proved to be both reliable and efficient.

scholarly journals Nonlinear Partial Differential Equations Arising from Prescribed Curvature Problems

2020 ◽  
pp. 84-101
Author(s):  
Man Chun Leung
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Scalar Curvature ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pde ◽  
Minkowski Problem ◽  
Partial Differential ◽  
Prescribed Curvature ◽  
Nirenberg Problem ◽  
Curvature Problem

We consider two prominent nonlinear partial differential equations (nonlinear PDE) linked to the prescribed curvature problems, namely, the Minkowski problem and the KazdanWarner/Nirenberg problem (prescribed scalar curvature problem). This article addresses some of the modern techniques in analysis used to draw out a number of the profound features in these equations.

scholarly journals Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Meku Ayalew ◽  
Gashu Gadisa Kiltu ◽  
Gemechis File Duressa
Keyword(s):  
Numerical Methods ◽  
Difference Equations ◽  
Numerical Scheme ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Maximum Absolute Errors

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.

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