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Journal of Applied Analysis
Vol. 7, No. 2 (2001), pp. 257–269
NONLINEAR NEUMANN PROBLEMS
I. FARAGÓ and J. KARÁTSON
Received November 27, 2000 and, in revised form, March 20, 2001
Abstract. We consider the numerical solution of quasilinear elliptic
Neumann problems. The basic difficulty is the non-injectivity of the
operator, which can be overcome by suitable factorization. We extend
the gradient-finiteelementmethod (GFEM), introduced earlier by the
authors for Dirichlet problems, to the Neumann problem. The algorithm
diffusion equation. To simplify the exposition, we consider only the case of polygonal/polyhedral domains and subdomains, and the case of homogeneous Neumann conditions. For the condensed matrix arising in mixed hybrid or mixed macro-hybrid finiteelementmethods we describe the general idea for construction of preconditioners with special projectors  . In Section 2 , we consider the model diffusion problem in the square domain partitioned in square subdomains with constant diffusion and reaction coefficients in subdomains. Based on the results of Section 1 we
. Appl. Math.
62 (2002), 870–887. http://dx.doi.org/10.1137/S0036139900375227  C. Li, Z. Zhao, and Y. Chen, Numerical approximation and error estimates of a time fractional order diffusion equation. In: Proc. of the ASME 2009 Intern. Design Engineering Technical Conf. and Computers and Information in Engineering Conf. IDETC/CIE 2009, San Diego, California, USA.  Y. Jiang and J. Ma, High-order finiteelementmethods for timefractional partial differential equations. J. Comput. Appl. Math.
1 Introduction The beginning of the analysis of the mixed finiteelementmethods for solving problems of the theories of plates, thin and medium thickness shells dates back to the 60s of the preceding century.
Among pioneering, papers [ 10 , 11 , 32 , 23 , 12 , 24 , 22 ] can be mentioned.
The research in this area is still very intensive.
See, for example, [ 28 ] and the articles cited therein. The presented work is devoted to the study of a mixed finiteelementmethod for geometrically nonlinear problems in the theory of thin elastic shells.
The method of
, Physical Review E 92 : 053202. Karczewska, A., Szczeciński, M., Rozmej, P., and Boguniewicz, B. (2016). Finiteelementmethod for stochastic extended KdV equations, Computational Methods in Science and Technology 22 (1): 19–29. Kim, J.W., Bai, K.J., Ertekin, R.C. and Webster, W.C. (2001). A derivation of the Green–Naghdi equations for irrotational flows, Journal of Engineering Mathematics 40 : 17–42. Marchant, T. and Smyth, N. (1990). The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, Journal of Fluid Mechanics 221 (1): 263
of the potential field changes from a pure Neumann condition on anodes and the insulated part to a nonlinear
one on cathodes. This causes local solution singularities around the interface between different parts of
the boundary [ 12 , 13 ]. Hence, the adaptive technique is a natural choice for its efficient numerical simulation. In this work, we develop and analyze an adaptive finiteelementmethod (AFEM) for
the system ( 1.1 )–( 1.2 ). A typical adaptive algorithm comprises successive iterations of the following
loop: (1.5) SOLVE → ESTIMATE → MARK
, International Journal for Numerical Methods in Engineering, 44 (9), 1267-1282, 1999. 5. J.C. Galvez, J. Cervenka, D.A. Cendon, V. Saouma, A discrete crack approach to normal/shearcracking of concrete, Cement and Concrete Research, 32 (10), 1567-1585, 2002. 6. F. Zhou, J.F. Molinari, Dynamic crack propagation with cohesive elements: a methodology to addressmesh dependency, International Journal for Numerical Methods in Engineering, 59 (1), 1-24, 2004. 7. T. Belytschko, J. Fish, B.E. Englemann, A finiteelementmethod with embedded localization zones, Computer Methods in