Discrete maximum principle for a 1D problem with piecewise-constant coefficients solved by hp-FEM

In this paper we prove the discrete maximum principle for a one-dimensional equation of the form –(au′)′ = f with piecewise-constant coefficient a(x), discretized by the hp-FEM. The discrete problem is transformed in such a way that the discontinuity of the coefficient a(x) disappears. Existing results are then applied to obtain a condition on the mesh which guarantees the satisfaction of the discrete maximum principle. Both Dirichlet and mixed Dirichlet–Neumann boundary conditions are discussed.

[1]  Todd F. Dupont,et al.  Failure of the discrete maximum principle for an elliptic finite element problem , 2004, Math. Comput..

[2]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[3]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[4]  Ansgar Jüngel,et al.  Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations , 2005, Numerische Mathematik.

[5]  Tom.  METHOD OF LINES AND CONSERVATION OF NONNEGATIVITY , 2004 .

[6]  Alexandre Ern,et al.  Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes , 2004 .

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

[8]  C. Schwab.  P- and hp- finite element methods : theory and applications in solid and fluid mechanics , 1998 .

[9]  E. G. Yanik.  Sufficient conditions for a discrete maximum principle for high order collocation methods , 1989 .

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

[11]  Tomáš Vejchodský,et al.  A weak discrete maximum principle for hp-FEM , 2007 .

[12]  A. H. Schatz,et al.  A weak discrete maximum principle and stability of the finite element method in _{∞} on plane polygonal domains. I , 1980 .

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

[14]  Tomaÿs Vejchodsky.  Discrete Maximum Principle for Poisson Equation with Mixed Boundary Conditions Solved by hp- FEM , 2006 .

[15]  I. Doležel,et al.  Higher-Order Finite Element Methods , 2003 .

[16]  Sergey Korotov,et al.  Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle , 2001, Math. Comput..

[17]  Sergey Korotov,et al.  Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions , 2005, Numerische Mathematik.

[18]  Tomáš Vejchodský,et al.  On the Nonnegativity Conservation in Semidiscrete Parabolic Problems , 2004 .

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

[20]  P. Solín.  Partial differential equations and the finite element method , 2005 .

[21]  Ivo Babuška,et al.  Approximation properties of the h-p version of the finite element method , 1996 .