TY - JOUR
T1 - A Coercivity Result of Quadratic Finite Volume Element Schemes over Triangular Meshes
AU - Wen , Xueying
AU - Zhou , Yanhui
JO - Advances in Applied Mathematics and Mechanics
VL - 4
SP - 901
EP - 931
PY - 2023
DA - 2023/04
SN - 15
DO - http://doi.org/10.4208/aamm.OA-2021-0311
UR - https://global-sci.org/intro/article_detail/aamm/21596.html
KW - Quadratic finite volume element schemes, triangular meshes, coercivity result, minimum angle condition.
AB - <p style="text-align: justify;">In this work, we study the coercivity of a family of quadratic finite volume element (FVE) schemes over triangular meshes for solving elliptic boundary value
problems. The analysis is based on the standard mapping from the trial function space
to the test function space so that the coercivity result can be naturally incorporated
with most existing theoretical results such as $H^1$ and $L^2$ error estimates. The novelty
of this paper is that, each element stiffness matrix of the quadratic FVE schemes can be
decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between
the FVE and FEM on the element boundary, while the third part can be expressed
as the tensor product of two vectors. As a result, we reach a sufficient condition to
guarantee the existence, uniqueness and coercivity result of the FVE solution on general triangular meshes. Moreover, based on this sufficient condition, some minimum
angle conditions with simple, analytic and computable expressions are obtained. By
comparison, the existing minimum angle conditions were obtained numerically from
a computer program. Theoretical findings are conformed with the numerical results.</p>