Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data

1987 ◽  
Vol 51 (5) ◽  
pp. 559-569 ◽  
Author(s):  
Hans-Peter Helfrich
Keyword(s):  
Initial Data ◽  
Error Estimates ◽  
Evolution Equations ◽  
Nonsmooth Initial Data ◽  
Semilinear Evolution Equations

Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data

2018 ◽  
Vol 52 (2) ◽  
pp. 773-801 ◽  
Author(s):  
Samir Karaa ◽  
Amiya K. Pani
Keyword(s):  
Initial Data ◽  
Error Estimate ◽  
Error Estimates ◽  
Fractional Order ◽  
Evolution Equations ◽  
Optimal Error Estimate ◽  
Backward Euler ◽  
Liouville Fractional Derivative ◽  
Nonsmooth Initial Data ◽  
Numer Anal

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v  ∈  L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

scholarly journals Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun
Keyword(s):  
Evolution Equations ◽  
Global Solutions ◽  
Global Attractors ◽  
Nonlinear Evolution Equations ◽  
Time Behavior ◽  
Asymptotic Regularity ◽  
Compact Attractor ◽  
Long Time ◽  
Critical Exponential Growth ◽  
Semilinear Evolution Equations

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.

scholarly journals Some Time Stepping Methods for Fractional Diffusion Problems with Nonsmooth Data

2018 ◽  
Vol 18 (1) ◽  
pp. 129-146 ◽  
Author(s):  
Yan Yang ◽  
Yubin Yan ◽  
Neville J. Ford
Keyword(s):  
Initial Data ◽  
Convergence Rate ◽  
Homogeneous Problem ◽  
Fractional Diffusion ◽  
Nonsmooth Data ◽  
Time Stepping ◽  
Time Discretisation ◽  
Nonsmooth Initial Data ◽  
Densely Defined ◽  
Diffusion Problems

AbstractWe consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha [18] established an {O(k)} convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator A is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where k denotes the time step size. In this paper, we approximate the Riemann–Liouville fractional derivative by Diethelm’s method (or L1 scheme) and obtain the same time discretisation scheme as in McLean and Mustapha [18]. We first prove that this scheme has also convergence rate {O(k)} with nonsmooth initial data for the homogeneous problem when A is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretisation scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth initial data. Using this new time discretisation scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is {O(k^{1+\alpha})}, {0<\alpha<1}, with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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