Discrete maximum principles for FEM solutions of some nonlinear elliptic interface problems

Discrete maximum principles are proved for flnite element dis- cretizations of nonlinear elliptic interface problems with jumps of the normal derivatives. The geometric conditions in the case of simplicial meshes are suit- able acuteness or nonobtuseness properties.

[1]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

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

[3]  Habib Ammari,et al.  Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter , 2002 .

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

[5]  Lubin G. Vulkov,et al.  Analysis of Immersed Interface Difference Schemes for Reaction-diffusion Problems with Singular Own Sources , 2003 .

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

[7]  T. D. Mast,et al.  Focusing and imaging using eigenfunctions of the scattering operator. , 1997, The Journal of the Acoustical Society of America.

[8]  H. Ammari,et al.  Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations , 2001 .

[9]  Michael Vogelius,et al.  A backprojection algorithm for electrical impedance imaging , 1990 .

[10]  Sol ´ õn.  On a Weak Discrete Maximum Principle for hp-FEM , 2006 .

[11]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[12]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[13]  Lubin G. Vulkov.  Well Posedness and a Monotone Iterative Method for a Nonlinear Interface Problem on Disjoint Intervals , 2007 .

[14]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[15]  Ekaterina Iakovleva,et al.  MUSIC-Type Electromagnetic Imaging of a Collection of Small Three-Dimensional Inclusions , 2007, SIAM J. Sci. Comput..

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  Kazufumi Ito,et al.  Maximum Principle Preserving Schemes for Interface Problems with Discontinuous Coefficients , 2001, SIAM J. Sci. Comput..

[18]  Mefire.  NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS , 2008 .

[19]  M. Krízek,et al.  Acute Versus Nonobtuse Tetrahedralizations , 2004 .

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

[21]  Three-dimensional computation of a magnetic field by mixed finite elements and boundary elements , 2000 .

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

[23]  Sergey Korotov,et al.  Global and local refinement techniques yielding nonobtuse tetrahedral partitions , 2005 .

[24]  Sergey Korotov,et al.  Finite element analysis of varitional crimes for a quasilinear elliptic problem in 3D , 2000, Numerische Mathematik.

[25]  M. Hanke,et al.  Numerical implementation of two noniterative methods for locating inclusions by impedance tomography , 2000 .

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

[27]  Kazuo Ishihara,et al.  Strong and weak discrete maximum principles for matrices associated with elliptic problems , 1987 .

[28]  Sergey Korotov,et al.  The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem , 2008 .

[29]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

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

[31]  D. Dobson,et al.  An image-enhancement technique for electrical impedance tomography , 1994 .

[32]  Juri D. Kandilarov.  A Monotone Iterative Method for Numerical Solution of Diffusion Equations with Nonlinear Localized Chemical Reactions , 2006, Numerical Methods and Applications.

[33]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[34]  János Karátson,et al.  Mesh independent superlinear convergence of an inner-outer iterative method for semilinear elliptic interface problems , 2009 .

[35]  Pekka Neittaanmäki,et al.  Mathematical and Numerical Modelling in Electrical Engineering Theory and Applications , 1996 .

[36]  Sergey Korotov,et al.  On discrete maximum principles for nonlinear elliptic problems , 2007, Math. Comput. Simul..

[37]  Endre Süli,et al.  Stability of the Lagrange-Galerkin method with non-exact integration , 1988 .

[38]  A minimum principle for superharmonic functions subject to interface conditions , 1981 .

[39]  David Isaacson,et al.  Layer stripping: a direct numerical method for impedance imaging , 1991 .

[40]  Sergey Korotov,et al.  Dissection of the path-simplex in R-n into n path-subsimplices , 2007 .

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

[42]  Sergey Korotov,et al.  Acute Type Refinements of Tetrahedral Partitions of Polyhedral Domains , 2001, SIAM J. Numer. Anal..

[43]  Ekaterina Iakovleva,et al.  A MUSIC Algorithm for Locating Small Inclusions Buried in a Half-Space from the Scattering Amplitude at a Fixed Frequency , 2005, Multiscale Model. Simul..

[44]  Darko Volkov,et al.  Numerical methods for locating small dielectric inhomogeneities , 2003 .

[45]  Robert V. Kohn,et al.  Numerical implementation of a variational method for electrical impedance tomography , 1990 .

[46]  H. Ammari,et al.  Reconstruction of Small Inhomogeneities from Boundary Measurements , 2005 .

[47]  Enrico Bertolazzi,et al.  A Second-Order Maximum Principle Preserving Finite Volume Method for Steady Convection-Diffusion Problems , 2005, SIAM J. Numer. Anal..

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

[49]  Michael Vogelius,et al.  Identification of conductivity imperfections of small diameter by boundary measurements. Continuous , 1998 .

[50]  O. A. Ladyzhenskai︠a︡,et al.  Linear and quasilinear elliptic equations , 1968 .

[51]  Thomas Y. Hou,et al.  Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients , 1999, Math. Comput..

[52]  István Faragó,et al.  Numerical solution of nonlinear elliptic problems via preconditioning operators : theory and applications , 2002 .

[53]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[54]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .