A finite element collocation method for bisingular integral equations. (English) Zbl 0548.65096
The convergence of a finite-element-collocation method is proved for bisingular integral equations with continuous coefficients on the torus. It is considered a special \(C^*\)-algebra of projection operators closely related to the collocation method for singular integral equations with continuous coefficients on the unit circle. It turns out that this algebra is isometrically isomorphic to the algebra of all continuous functions on a cylinder. This fact permits a very ”sensitive” localization. The convergence of the collocation method is proved for bisingular equations by means of local principles and tensor product arguments.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E05 | Integral equations with kernels of Cauchy type |
Keywords:
finite-element-collocation; bisingular integral equations; \(C^*\)-algebra of projection operatorsReferences:
| [1] | Böttcher A, Math. Nachr. 110 pp 279– (1983) · Zbl 0549.47010 · doi:10.1002/mana.19831100120 |
| [2] | Dixmier P. J., Les C-Algebres ET Leurs Representations (1969) |
| [3] | Gelfand I. M., Kommutative normierte Algebren (1966) |
| [4] | Gohberg I. C., Einführung in die Theorie der eindimensional en singularen Integral operatoren (1979) · doi:10.1007/978-3-0348-5555-6 |
| [5] | Gohberg I. C., Funkc. anal. 3 (2) pp 46– (1969) |
| [6] | Gohberg I. C., Funkc. anal. 4 (3) pp 26– (1970) |
| [7] | Junghanns P., Prepr. d. Akad. d. Wiss. d. DDR |
| [8] | Kozak A. V., DAN SSSR 212 6 (3) pp 1287– (1973) |
| [9] | Pilidi V. S., Matem. issled., Kisinev, t. 7, v. 3 (25) pp 167– (1972) |
| [10] | Prü{\(\beta\)}dorf S, Teubner-Texte zur Math. Probleme und Methoden der Math. Physik |
| [11] | Prü{\(\beta\)}dorf S, Integral Equations and Operator Theory |
| [12] | Prü{\(\beta\)}dorf S., Math. Nachr. 100 (25) pp 33– (1981) |
| [13] | Prö{\(\beta\)}dorf S., Prepr. d. Akad. d. Wiss. d. DDR, |
| [14] | Silbeimaan B., ZfAA, Bd. 1 (6) pp 45– (1982) |
| [15] | Silbermaaa B., Math. Nachr. 104 (6) pp 137– (1981) · Zbl 0494.47018 · doi:10.1002/mana.19811040111 |
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