TY - JOUR
T1 - On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian
AU - Cross , Logan J.
AU - Zhang , Xiangxiong
JO - Annals of Applied Mathematics
VL - 2
SP - 161
EP - 190
PY - 2024
DA - 2024/05
SN - 40
DO - http://doi.org/10.4208/aam.OA-2024-0007
UR - https://global-sci.org/intro/article_detail/aam/23099.html
KW - Inverse positivity, discrete maximum principle, high order accuracy, monotonicity, discrete Laplacian, spectral element method.
AB - <p style="text-align: justify;">The monotonicity of discrete Laplacian, i.e., inverse positivity of
stiffness matrix, implies discrete maximum principle, which is in general not true
for high order accurate schemes on unstructured meshes. On the other hand,
it is possible to construct high order accurate monotone schemes on structured
meshes. All previously known high order accurate inverse positive schemes are
or can be regarded as fourth order accurate finite difference schemes, which is
either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element
method for the two-dimensional Laplacian, we prove its stiffness matrix is a
product of four M-matrices thus it is unconditionally monotone. Such a scheme
can be regarded as a fifth order accurate finite difference scheme. Numerical tests
suggest that the unconditional monotonicity of $Q^k$ spectral element methods will
be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other
words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral
element methods can be unconditionally monotone in three dimensions.</p>