Full Discretisation of Second-order Nonlinear Evolution Equations: Strong Convergence and Applications

Emmrich, Etienne; Šiška, David
October 2011
Computational Methods in Applied Mathematics;2011, Vol. 11 Issue 4, p441
Academic Journal
Recent results on convergence of fully discrete approximations combining the Galerkin method with the explicit-implicit Euler scheme are extended to strong convergence under additional monotonicity assumptions. It is shown that these abstract results, formulated in the setting of evolution equations, apply, for example, to the partial differential equation for vibrating membrane with nonlinear damping and to another partial differential equation that is similar to one of the equations used to describe martensitic transformations in shape-memory alloys. Numerical experiments are performed for the vibrating membrane equation with nonlinear damping which support the convergence results.


Related Articles

  • A numerical approach to study the properties of solutions of the diffusive wave approximation of the shallow water equations. Mauricio Santillana; Clint Dawson // Computational Geosciences;Jan2010, Vol. 14 Issue 1, p31 

    Abstract  In this paper, we study the properties of approximate solutions to a doubly nonlinear and degenerate diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW), using a numerical approach based on the Galerkin...

  • Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems. Alkan, Sertan; Secer, Aydin // Mathematical Problems in Engineering;2/25/2015, Vol. 2015, p1 

    We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions...

  • A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation. Chan, Hiu Ning; Chung, Eric T. // Numerical Mathematics: Theory, Methods & Applications;Nov2015, Vol. 8 Issue 4, p451 

    The staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied...

  • Fourth-Order Splitting Methods for Time-Dependant Differential Equations. Geiser, Jürgen // Numerical Mathematics: Theory, Methods & Applications;Aug2008, Vol. 1 Issue 3, p321 

    This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating...

  • Hyperbolic diffusion in chaotic systems. Borys, P.; Grzywna, Z.; Łuczka, J. // European Physical Journal B -- Condensed Matter;Sep2011, Vol. 83 Issue 2, p223 

    We consider a deterministic process described by a discrete one-dimensional chaotic map and study its diffusive-like properties. Starting with the corresponding Frobenius-Perron equation we derive an approximate evolution equation for the probability distribution which is a partial differential...

  • The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations. Caraballo, T.; Kloeden, P. E. // Applied Mathematics & Optimization;Nov/Dec2006, Vol. 54 Issue 3, p401 

    Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin...

  • A smoothed Hermite radial point interpolation method for thin plate analysis. Cui, Xiangyang; Liu, Guirong; Li, Guangyao // Archive of Applied Mechanics;Jan2011, Vol. 81 Issue 1, p1 

    smoothed Hermite radial point interpolation method using gradient smoothing operation is formulated for thin plate analysis. The radial basis functions augmented with polynomial basis are used to construct the shape functions that have the important Delta function property. The smoothed Galerkin...

  • Numerical Solution and Simulation of Second-Order Parabolic PDEs with Sinc-Galerkin Method Using Maple. Secer, Aydin // Abstract & Applied Analysis;2013, p1 

    An efficient solution algorithm for sinc-Galerkin method has been presented for obtaining numerical solution of PDEs with Dirichlet-type boundary conditions by using Maple Computer Algebra System. The method is based on Whittaker cardinal function and uses approximating basis functions and their...

  • The evolution partial differential equation ut=uxxx+3(uxxu2 +3u2xu)+3uxu4. Calogero, F. // Journal of Mathematical Physics;Mar1987, Vol. 28 Issue 3, p538 

    The evolution equation ut=uxxx+3(uxxu2 +3u2xu)+3uxu4, u=u(x,t), is integrable; it can be (exactly) linearized by an appropriate change of (dependent) variable. Hence several explicit solutions of the partial differential equation (PDE) can be exhibited; some of them display a remarkable...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics