Error estimates over infinite intervals of some discretizations of evolution equations

1984 ◽  
Vol 24 (4) ◽  
pp. 413-424 ◽  
Author(s):  
O. Axelsson
Keyword(s):  
Error Estimates ◽  
Evolution Equations ◽  
Infinite Intervals

Error estimates of a finite element method for stochastic time-fractional evolution equations with fractional Brownian motion

2021 ◽  
pp. 2250006
Author(s):  
Jingyun Lv
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Error Estimates ◽  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Backward Euler ◽  
Stochastic Time ◽  
Element Method

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional Brownian motion. The spatial and temporal regularity of the mild solution is given. The numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula, and the noise by the [Formula: see text]-projection. The strong convergence error estimates for both semi-discrete and fully discrete schemes are established. A numerical example is presented to verify our theoretical analysis.

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.

Sign in / Sign up

Export Citation Format

Share Document