On the monotonicity of high order discrete Laplacian

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, im5 plies discrete maximum principle, which is in general not true for high order schemes on unstructured meshes. But on structured meshes, it is possible to have high order accurate monotone schemes. We first review previously known high order accurate inverse positive schemes, all of which are fourth order accurate with proven monotonicity on uniform meshes. Then we discuss the monotonicity of a fourth order variational difference scheme on quasi-uniform meshes and prove the inverse positivity of a fifth order accurate variational difference scheme on a uniform mesh.

[1]  J. Lorenz,et al.  Zur Inversmonotonie diskreter Probleme , 1977 .

[2]  Tomás Vejchodský,et al.  Discrete maximum principle for higher-order finite elements in 1D , 2007, Math. Comput..

[3]  L. Fox,et al.  Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations , 1947, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  Ludmil T. Zikatanov,et al.  A monotone finite element scheme for convection-diffusion equations , 1999, Math. Comput..

[5]  R. Plemmons.  M-matrix characterizations.I—nonsingular M-matrices , 1977 .

[6]  Shusen Xie,et al.  A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Scalar Convection Diffusion Equations , 2018, SIAM J. Numer. Anal..

[7]  Philippe G. Ciarlet,et al.  Discrete maximum principle for finite-difference operators , 1970 .

[8]  B. E. Hubbard,et al.  On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation , 1962 .

[9]  L. Wahlbin.  Superconvergence in Galerkin Finite Element Methods , 1995 .

[10]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[11]  L. Kantorovich,et al.  Approximate methods of higher analysis , 1960 .

[12]  Hans D. Mittelmann,et al.  Some remarks on the discrete maximum-principle for finite elements of higher order , 1981, Computing.

[13]  Hao Li,et al.  Superconvergence of High Order Finite Difference Schemes Based on Variational Formulation for Elliptic Equations , 2019, J. Sci. Comput..

[14]  J. Bramble,et al.  New monotone type approximations for elliptic problems , 1964 .

[15]  LAGRANGIAN FINITE ELEMENT and FINITE DIFFERENCE METHODS FOR POISSON PROBLEMS. , 1975 .

[16]  Xiangxiong Zhang,et al.  On the monotonicity and discrete maximum principle of the finite difference implementation of \(C^0\) - \(Q^2\) finite element method , 2019, Numerische Mathematik.

[17]  J. Bramble,et al.  On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type , 1964 .

[18]  Bengt Fornberg,et al.  A primer on radial basis functions with applications to the geosciences , 2015, CBMS-NSF regional conference series in applied mathematics.

[19]  Jens Lorenz,et al.  Inverse monotonicity and difference schemes of higher order. A summary for two-point boundary value problems , 1979 .

[20]  L. Collatz.  The numerical treatment of differential equations , 1961 .

[21]  Jinchao Xu,et al.  Superconvergence of quadratic finite elements on mildly structured grids , 2008, Math. Comput..

[22]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[23]  Charles A. Hall,et al.  The maximum principle for bilinear elements , 1984 .

[24]  James H. Bramble,et al.  Fourth-order finite difference analogues of the Dirichlet problem for Poisson’s equation in three and four dimensions , 1963 .