Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method

A boundary integral equation (BIE) formulation is presented for the numerical solution of certain two dimensional nonlinear elliptic equations subject to nonlinear boundary conditions. By applying the Kirchoff transformation, all nonlinear aspects are first transferred to the boundary of the solution domain. Then the accurate solution of problems in which there are boundary singularities is demonstrated by including the analytic nature of the singular solution in only those regions nearest the singularity. Because of this, only a minor modification of the classical nonlinear BIE is required and this results in a substantial improvement in the accuracy of the numerical results throughout the entire solution domain.

[1]  T. Patterson,et al.  The optimum addition of points to quadrature formulae. , 1968 .

[2]  J. Whiteman,et al.  Numerical solution of two dimensional harmonic boundary problems containing singularities by conformal transformation methods , 1971 .

[3]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[4]  Andrzej J. Nowak,et al.  Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions , 1981 .

[5]  C. Atkinson,et al.  The treatment of singularities in the calculation of stress intensity factors using the boundary integral equation method , 1981 .

[6]  Graeme Fairweather,et al.  On the numerical solution of two-dimensional potential problems by an improved boundary integral equation method☆ , 1979 .

[7]  M. A. Jaswon,et al.  Integral equation methods in potential theory and elastostatics , 1977 .

[8]  D. Ingham,et al.  The Numerical Solution of Plane Potential Problems by Improved Boundary Integral Equation Methods , 1981 .

[9]  H. Motz,et al.  The treatment of singularities of partial differential equations by relaxation methods , 1947 .

[10]  Mark A. Kelmanson,et al.  Modified Integral Equation Solution of Viscous Flows Near Sharp Corners , 1983 .

[11]  F. Kreith,et al.  Principles of heat transfer , 1962 .

[12]  Lee W. Johnson,et al.  Numerical Analysis , 1977 .

[13]  M. Hanna,et al.  An iterative boundary integral numerical solution for general steady heat conduction problems , 1981 .

[14]  L. C. Woods.  THE RELAXATION TREATMENT OF SINGULAR POINTS IN POISSON'S EQUATION , 1953 .

[15]  C. Brebbia,et al.  Boundary Element Techniques in Engineering , 1979 .

[16]  Derek B. Ingham,et al.  Boundary Integral Equation Analysis of Transmission-Line Singularities , 1981 .

[17]  Derek B. Ingham,et al.  Boundary Integral Equation Solution of Non-linear Plane Potential Problems , 1981 .