ChesterRep Collection:
http://hdl.handle.net/10034/6981
Tue, 14 Feb 2017 02:59:09 GMT2017-02-14T02:59:09ZBlending low-order stabilised finite element methods: a positivity preserving local projection method for the convection-diffusion equation
http://hdl.handle.net/10034/620327
Title: Blending low-order stabilised finite element methods: a positivity preserving local projection method for the convection-diffusion equation
Authors: Barrenechea, Gabriel; Burman, Erik; Karakatsani, Fotini
Abstract: In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal
order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.Fri, 20 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6203272017-01-20T00:00:00ZHigh-Order Numerical Methods for Solving Time Fractional Partial Differential Equations
http://hdl.handle.net/10034/620273
Title: High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations
Authors: Li, Zhiqiang; Liang, Zongqi; Yan, Yubin
Abstract: In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0<α<1
O(τ^(3−α)+h^2),0<α<1
are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where τ and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.
Description: The final publication is available at Springer via http://dx.doi.org/10.1007/s10915-016-0319-1Tue, 15 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10034/6202732016-11-15T00:00:00ZThe sharp interface limit for the stochastic Cahn-Hilliard Equation
http://hdl.handle.net/10034/620253
Title: The sharp interface limit for the stochastic Cahn-Hilliard Equation
Authors: Antonopoulou, Dimitra; Bloemker, Dirk; Karali, Georgia
Abstract: We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp
interface limit, where the positive parameter \eps tends to zero, which measures the width of transition
layers generated during phase separation. We also couple the noise strength to this parameter.
Using formal asymptotic expansions, we identify the limit. In the right scaling we indicate that the
solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic
forcing. In the case when the noise is sufficiently small, we rigorously prove that the limit is a
deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to
extend the rigorous result to larger noise strength.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6202532017-01-01T00:00:00ZOn a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics
http://hdl.handle.net/10034/620247
Title: On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics
Authors: Kavallaris, Nikos I.; Lankeit, Johannes; Winkler, Michael
Abstract: We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for
\[
u_t = u \Delta u + u \int_\Omega |\nabla u|^2
\]
in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. In particular, it is shown that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global.Tue, 01 Nov 2016 00:00:00 GMThttp://hdl.handle.net/10034/6202472016-11-01T00:00:00Z