Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method

Title:

Some new error estimates for finite element methods for second order hyperbolic equations using the Newmark method

Creator:

Abdallah and Fuhrmann, Jürgen

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:ac51969a-3605-4b6b-8d31-69d3e952f85d
uuid:ac51969a-3605-4b6b-8d31-69d3e952f85d
doi:10.21136/MB.2014.143843

Subject:

acoustic wave equation, finite element method, Newmark method, and new error estimate

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

We consider a family of conforming finite element schemes with piecewise polynomial space of degree k in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is h k + τ 2 in the discrete norms of L∞(0, T ; H1 (Ω)) and W1,∞(0, T ; L 2 (Ω)), where h and τ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

125-136

Source:

Mathematica bohemica | 2014 Volume:139 | Number:2

Harvested from:

CDK

Metadata only:

false