An $$\alpha $$-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative

2021 ◽  
Vol 86 (2) ◽  
Author(s):  
Kim-Ngan Le ◽  
Martin Stynes
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Order ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

2020 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inner Product ◽  
Polynomial Degree ◽  
Finite Element Approximations ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Abstract For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 815-825 ◽  
Author(s):  
Chaobao Huang ◽  
Xiaohui Liu ◽  
Xiangyun Meng ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Value Problem ◽  
Difference Method ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Graded Meshes ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is {(0,1)^{d}} for some {d\in\{1,2,3\}}. This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

scholarly journals On Temporal Behaviour of Solutions in Thermoelasticity of Porous Micropolar Bodies

2014 ◽  
Vol 22 (1) ◽  
pp. 169-188 ◽  
Author(s):  
Marin Marin ◽  
Olivia Florea
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cesàro Means ◽  
Temporal Behaviour ◽  
Cesáro Means ◽  
Thermoelastic Body ◽  
Initial Boundary Value

AbstractWe consider a porous thermoelastic body, including voidage time derivative among the independent constitutive variables. For the initial boundary value problem of such materials, we analyze the temporal behaviour of the solutions. To this aim we use the Cesaro means for the components of energy and prove the asymptotic equipartition in mean of the kinetic and strain energies.

scholarly journals WELL-POSED INITIAL-BOUNDARY VALUE PROBLEM FOR A CONSTRAINED EVOLUTION SYSTEM AND RADIATION-CONTROLLING CONSTRAINT-PRESERVING BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 587-612 ◽  
Author(s):  
ALEXANDER M. ALEKSEENKO
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Evolution System ◽  
Constrained Evolution ◽  
Initial Boundary Value ◽  
Well Posed

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e. the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier–Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.

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