TY - JOUR
T1 - A Finite Volume Element Solution Based on Postprocessing Technique over Arbitrary Convex Polygonal Meshes
AU - Zhang , Yanlong
AU - Zhou , Yanhui
JO - International Journal of Numerical Analysis and Modeling
VL - 5
SP - 597
EP - 617
PY - 2023
DA - 2023/09
SN - 20
DO - http://doi.org/10.4208/ijnam2023-1026
UR - https://global-sci.org/intro/article_detail/ijnam/22004.html
KW - Finite volume element solution, postprocessing technique, convex polygonal meshes, existence and uniqueness, $H^1$ and $L^2$ error estimates.
AB - <p style="text-align: justify;">A special finite volume element method based on postprocessing technique is proposed
to solve the anisotropic diffusion problem on arbitrary convex polygonal meshes. The shape
function of polygonal finite element method is constructed by Wachspress generalized barycentric
coordinate, and by adding some element-wise bubble functions to the finite element solution, we
get a new finite volume element solution that satisfies the local conservation law on a certain dual
mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each
primary cell, so that it is easy to implement. More interesting is that, a general construction of
the bubble functions is introduced on each polygonal cell, which enables us to prove the existence
and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full
anisotropic diffusion tensor. The optimal $H^1$ and $L^2$ error estimates of the post-processed solution
are also obtained. Finally, the local conservation property and convergence of the new polygonal
finite volume element solution are verified by numerical experiments.</p>