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| On the Monotonicity of Q3 SpectralElement Method for Laplacian |
| Logan J. Cross,Xiangxiong Zhang |
| (Department of Mathematics, Purdue University, 150 N. University Street,
West Lafayette, IN 47907-2067, USA) |
| DOI： |
| Abstract: |
| 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 Q3
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 Qk
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 Q2 and Q3
spectral
element methods can be unconditionally monotone in three dimensions. |
| Key words: Inverse positivity, discrete maximum principle, high order accuracy, monotonicity, discrete Laplacian, spectral element method. |