A finite element method for some integral equations of the first kind. (English) Zbl 0352.45016


45L05 Theoretical approximation of solutions to integral equations
49M15 Newton-type methods
45B05 Fredholm integral equations
47Gxx Integral, integro-differential, and pseudodifferential operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
Full Text: DOI


[1] Anselone, P.M, Collectively compact operator approximation theory, (1971), Prentice Hall London · Zbl 0228.47001
[2] Atkinson, K.A, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, () · Zbl 0353.65069
[3] Aubin, J.P, Approximation of elliptic boundary-value problems, (1972), Wiley-Interscience New York
[4] Bramble, J; Schatz, A, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Commun. pure appl. math., 23, 653-675, (1970) · Zbl 0204.11102
[5] Brunner, H, The solution of Volterra integral equations of the first kind by piecewise polynomials, J. inst. math. appl., 12, 295-302, (1973) · Zbl 0271.65066
[6] {\scS. Christiansen}, Integral equations without a unique solution can be made useful for solving some plane harmonic problems, J. Inst. Math. Appl., to appear. · Zbl 0309.45001
[7] Christiansen, S; Rasmussen, H, Numerical solution for two-dimensional annular electrochemical machining problems, ()
[8] Courant, R; Hilbert, D, ()
[9] Cyrlin, L.E; Cyrlin, L.E, On a method of solving integral equations of the first kind in potential theory problems, Z. vycisl. mat. i mat. fiz., USSR comput. phys., 9, 324-328, (1969), translated in · Zbl 0181.19004
[10] Fichera, G, Linear elliptic equations of higher order in two independent variables and singular integral equations, () · Zbl 0111.29602
[11] Gilbert, R.P; Hsiao, G.C; Gilbert, R.P; Hsiao, G.C, On Dirichlet’s problem for quasilinear elliptic equations, (), 184-236 · Zbl 0338.35057
[12] Howland, J, Symmetrizing kernels and the integral equations of first kind of classical potential theory, (), 1-7 · Zbl 0164.42004
[13] Howland, J.L; Vaillancourt, R, Series expansions of induced potentials, J. math. anal. appl., 16, 385-395, (1966) · Zbl 0167.40202
[14] Hsiao, G.C; MacCamy, R.C, Solution of boundary value problems by integral equations of the first kind, SIAM rev., 15, 687-705, (1973) · Zbl 0235.45006
[15] Jaswon, M.A, Integral equation methods in potential theory I, (), 23-32 · Zbl 0112.33103
[16] Jaswon, M.A; Maiti, M; Symm, G.T, Numerical biharmonic analysis and some applications, Int. J. solids struct., 3, 309-322, (1967) · Zbl 0148.19403
[17] Kammerer, W.J; Nashed, M.Z, Iterative methods for best approximate solutions of linear integral equations of the first and second kinds, J. math. anal. appl., 40, 547-574, (1972) · Zbl 0246.45015
[18] Landweber, L, An iteration formula for Fredholm integral equations of the first kind, Amer. J. math. LXXIII, 615-624, (1951) · Zbl 0043.10602
[19] Le Roux, M.N, Résolution numérique du problème du potential dans le plan par une méthode variationelle d’éléments finis, Thèse, L’université de Rennes, Série A, no. D’ordre 347, no. de série 38, (1974)
[20] Le Roux, M.N, Équations intégrales pour le problème du potential électrique dans le plan, C. R. acad. sc., ser. A, 278, (1974)
[21] {\scM. N. Le Roux}, Méthode d’élément finis pour la résolution numérique de problèmes extérieurs en dimansion deux, to appear in R.A.I.R.O. · Zbl 0382.65055
[22] Lions, J.L; Magenes, E, ()
[23] MacCamy, R.C, On singular integral equations with logarithmic or Cauchy kernels, J. math. mech., 7, 355-376, (1958) · Zbl 0084.32201
[24] Muskhelishvili, N.L, Singular integral equations, (1953), Noordhoff Groningen, The Netherlands
[25] Nashed, M.Z; Nashed, M.Z, Approximate regularized solutions to improperly posed linear integral and operator equations, (), 289-332 · Zbl 0647.47013
[26] {\scJ. C. Nedelec}, Méthodes d’éléments finis courbés pour la résolution des équations intégrais singulières sur des surfaces de R3. 1-31, to appear in R.A.I.R.O.
[27] Nedelec, J.C; Planchard, J, Une méthode variationelle d’éléments finis pour la résolution numérique d’un problème extérieurs dans R3, Revue française d’automatique, informatique et recherche opérationnelle, no., 105-129, (décembre 1973), R3
[28] Nitsche, J, Verfahren von Ritz und spline-interpolation bei Sturm-Liouville-randwertproblemen, Numer. math., 13, 260-265, (1969) · Zbl 0181.18204
[29] Nitsche, J, Umkehrsätze für spline-approximationen, Composition math., 21, 400-416, (1969) · Zbl 0199.39302
[30] Nitsche, J, Zur konvergenz von Näherungsverfahren bezüglich verschiedener normen, Numer. math., 15, 224-228, (1970) · Zbl 0221.65092
[31] Noble, Ben, Error analysis of colocation methods for solving Fredholm integral equations, () · Zbl 0311.65068
[32] {\scBen Noble}, A bibliography on “Methods for Solving Integral Equations,” University of Wisconsin, Madison. · Zbl 0311.65068
[33] Schmeidler, W, Integralgleichungen mit anwendungen in physik und technik I, (1950), AVG Leipzig · Zbl 0035.34901
[34] Seeley, R, Topics in pseudo differential operators, (), 169-305
[35] {\scH. Siddalingaiah}, A constructive solution of the induced potential problem, SIAM J. Math. Anal, to appear.
[36] Stakgold, I, Boundary value problems of mathematical physics II, (1968), Macmillan New York · Zbl 0165.36803
[37] Symm, G.T, Integral equation methods in potential theory II, (), 33-46 · Zbl 0112.33201
[38] Symm, G.T, An integral equation method in conformal mapping, Numer. math., 9, 250-258, (1966) · Zbl 0156.16901
[39] Taylor, A.E, Functional analysis, (1958), Wiley New York
[40] Triebel, H, Höhere analysis, (1972), Deutscher Verlag d. Wiss Berlin · Zbl 0257.47001
[41] Warschawski, S.E, On the solution of the lichtenstein-gershgorin integral equation in conformal mapping: I theory, Nat. bur. stand. appl. math. ser., 42, 7-29, (1955)
[42] Wendland, W, Bemerkungen über die fredholmschen Sätze, Methoden und verf. der math. physik 3, 141-176, (1970), B. I. Mannheim 722 · Zbl 0211.16704
[43] Wendland, W; Wendland, W, An integral equation method for generalized analytic functions, (), 414-452
[44] Werner, H; Schaback, R, Praktische Mathematik II, (1972), Springer Berlin
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