TY - JOUR
T1 - A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number
AU - Du , Yu
AU - Zhang , Zhimin
JO - Communications in Computational Physics
VL - 1
SP - 133
EP - 156
PY - 2019
DA - 2019/10
SN - 22
DO - http://doi.org/ 10.4208/cicp.OA-2016-0121
UR - https://global-sci.org/intro/article_detail/cicp/13350.html
KW - Weak Galerkin finite element method, Helmholtz equation, large wave number, stability, error estimates.
AB - <p style="text-align: justify;">We 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 number k are derived. This shows that
the pollution error in the broken H<sup>1</sup>-norm is bounded by O(k(kh)<sup>2p</sup>) under mesh condition k<sup>7/2</sup>h<sup>2</sup> ≤C<sub>0</sub> or (kh)<sup>2</sup>+k(kh)<sup>p+1</sup> ≤C<sub>0</sub>, which coincides with the phase error of the
finite element method obtained by existent dispersion analyses. Here h is the mesh
size, p is the order of the approximation space and C<sub>0</sub> is a constant independent of k
and h. 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.</p>