scholarly journals Stability analysis and best approximation error estimates of discontinuous time-stepping schemes for the Allen–Cahn equation

2019 ◽  
Vol 53 (2) ◽  
pp. 551-583 ◽  
Author(s):  
Konstantinos Chrysafinos
Keyword(s):  
Error Estimates ◽  
Best Approximation ◽  
A Priori ◽  
Approximation Error ◽  
Gronwall Lemma ◽  
Discrete Approximations ◽  
Fully Discrete ◽  
Time Stepping ◽  
Cahn Equation ◽  
Boot Strap

Fully-discrete approximations of the Allen–Cahn equation are considered. In particular, we consider schemes of arbitrary order based on a discontinuous Galerkin (in time) approach combined with standard conforming finite elements (in space). We prove that these schemes are unconditionally stable under minimal regularity assumptions on the given data. We also prove best approximation a-priori error estimates, with constants depending polynomially upon (1/ε) by circumventing Gronwall Lemma arguments. The key feature of our approach is a carefully constructed duality argument, combined with a boot-strap technique.

ERROR ESTIMATES FOR LINEAR PDEs SOLVED BY WAVELET BASED TAYLOR–GALERKIN SCHEMES

2009 ◽  
Vol 07 (02) ◽  
pp. 143-162 ◽  
Author(s):  
MANI MEHRA ◽  
B. V. RATHISH KUMAR
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Equivalent Norm ◽  
Posteriori Error ◽  
Discrete Problem ◽  
Fully Discrete ◽  
Time Stepping ◽  
Priori Estimates ◽  
Space Error

In this paper, we develop a priori and a posteriori error estimates for wavelet-Taylor–Galerkin schemes introduced in Refs. 6 and 7 (particularly wavelet Taylor–Galerkin scheme based on Crank–Nicolson time stepping). We proceed in two steps. In the first step, we construct the priori estimates for the fully discrete problem. In the second step, we construct error indicators for posteriori estimates with respect to both time and space approximations in order to use adaptive time steps and wavelet adaptivity. The space error indicator is computed using the equivalent norm expressed in terms of wavelet coefficients.

scholarly journals ERROR ESTIMATES FOR TIME ACCURATE WAVELET BASED SCHEMES FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

2007 ◽  
Vol 05 (04) ◽  
pp. 667-678 ◽  
Author(s):  
MANI MEHRA ◽  
B. V. RATHISH KUMAR
Keyword(s):  
Error Estimates ◽  
A Priori Estimates ◽  
A Priori ◽  
Hyperbolic Partial Differential Equations ◽  
Discrete Problem ◽  
Spatial Error ◽  
Fully Discrete ◽  
Wavelet Approximation ◽  
Two Stages ◽  
Adjoint Operators

In this study, we derive error estimates for time accurate wavelet based schemes in two stages. First we look at the semi-discrete boundary value problem as a Cauchy problem and use spectral decomposition of self adjoint operators to arrive at the temporal error estimates both in L2 and energy norms. Later, following the wavelet approximation theory, we propose spatial error estimates in L2 and energy norms. And finally arrive at a priori estimates for the fully discrete problem.

scholarly journals A type III porous-thermo-elastic problem with quasi-static microvoids

Meccanica ◽  
2021 ◽  
Author(s):  
Noelia Bazarra ◽  
Alberto Castejón ◽  
José R. Fernández ◽  
Ramón Quintanilla
Keyword(s):  
Volume Fraction ◽  
A Priori ◽  
Linear Convergence ◽  
Euler Method ◽  
Point Of View ◽  
Discrete Approximations ◽  
Type Iii ◽  
Fully Discrete ◽  
Backward Euler ◽  
Additional Regularity

AbstractIn this work we study, from the numerical point of view, a one-dimensional thermoelastic problem where the thermal law is of type III. Quasi-static microvoids are also assumed within the model. The variational formulation leads to a coupled linear system made of variational equations and it is written in terms of the velocity, the volume fraction and the temperature. Fully discrete approximations are introduced by using the finite element method and the backward Euler method. A discrete stability property and a priori error estimates are proved, deriving the linear convergence under adequate additional regularity. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.

scholarly journals A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Yuping Zeng ◽  
Kunwen Wen ◽  
Fen Liang ◽  
Huijian Zhu
Keyword(s):  
Heat Equation ◽  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Fully Discrete ◽  
Backward Euler ◽  
Elliptic Reconstruction

We introduce and analyze a weakly overpenalized symmetric interior penalty method for solving the heat equation. We first provide optimal a priori error estimates in the energy norm for the fully discrete scheme with backward Euler time-stepping. In addition, we apply elliptic reconstruction techniques to derive a posteriori error estimators, which can be used to design adaptive algorithms. Finally, we present two numerical experiments to validate our theoretical analysis.

scholarly journals An adaptive time-stepping method based on a posteriori weak error analysis for large SDE systems

2021 ◽  
Author(s):  
Fabian Merle ◽  
Andreas Prohl
Keyword(s):  
A Priori ◽  
Approximation Error ◽  
Adaptive Method ◽  
Euler Method ◽  
Weak Approximation ◽  
A Posteriori ◽  
Time Stepping ◽  
Backward Equation ◽  
Adaptive Time ◽  
Weak Error

AbstractWe develop an adaptive algorithm for large SDE systems, which automatically selects (quasi-)deterministic time steps for the semi-implicit Euler method, based on an a posteriori weak error estimate. Main tools to construct the a posteriori estimator are the representation of the weak approximation error via Kolmogorov’s backward equation, a priori bounds for its solution and the Clark–Ocone formula. For a certain class of SDE systems, we validate optimal weak convergence order 1 of the a posteriori estimator, and termination of the adaptive method based on it within $${{\mathcal {O}}}(\mathtt{Tol}^{-1})$$ O ( Tol - 1 ) steps.

Numerical analysis of a piezoelectric bone remodelling problem

2012 ◽  
Vol 23 (5) ◽  
pp. 635-657 ◽  
Author(s):  
J. R. FERNÁNDEZ ◽  
J. M. GARCÍA-AZNAR ◽  
R. MARTÍNEZ
Keyword(s):  
Variational Formulation ◽  
Bone Remodelling ◽  
A Priori ◽  
Linear Convergence ◽  
Coupled System ◽  
Regularity Conditions ◽  
The Finite Element Method ◽  
Discrete Approximations ◽  
Fully Discrete ◽  
Additional Regularity

Although in recent years bone piezoelectricity has been normally neglected, lately a new interest has appeared to show the importance of bone piezoelectricity in wet bone's complex response to loading. Here we numerically study a problem, including a strain-adaptive bone remodelling and the piezoelectricity. Its variational formulation leads to a coupled system composed of two linear variational equations for displacements and electric potential, and a parabolic variational inequality for the apparent density. Fully discrete approximations are now introduced by using the finite element method to approximate spatial variable and the explicit Euler scheme to discretise time derivatives. Some a priori error estimates are proved and the linear convergence of the algorithm is deduced under additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are described to show the accuracy of the proposed algorithm and the behaviour of the solution.

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