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.

A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number

2017 ◽  
Vol 22 (1) ◽  
pp. 133-156 ◽  
Author(s):  
Yu Du ◽  
Zhimin Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Phase Error ◽  
Three Dimensions ◽  
Wave Number ◽  
Weak Galerkin ◽  
High Wave Number ◽  
Large Wave ◽  
Element Method

AbstractWe study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave numberkare derived. This shows that the pollution error in the brokenH1-norm is bounded byunder mesh conditionk7/2h2≤C0or (kh)2+k(kh)p+1≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Herehis the mesh size,pis the order of the approximation space andC0is a constant independent ofkandh. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

Nonconforming Mixed Finite Element Method for Time-dependent Maxwell's Equations with ABC

2016 ◽  
Vol 9 (2) ◽  
pp. 193-214
Author(s):  
Changhui Yao ◽  
Dongyang Shi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Absorbing Boundary Conditions ◽  
Time Dependent ◽  
Backward Euler ◽  
Element Method

AbstractIn this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semidiscrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.

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