On the convergence of the -version of the boundary element Galerkin method

Authors:
E. P. Stephan and M. Suri

Journal:
Math. Comp. **52** (1989), 31-48

MSC:
Primary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-1989-0947469-5

MathSciNet review:
947469

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove convergence for the *p*-version of Galerkin boundary element schemes applied to various first-kind integral equations. We establish optimal error estimates for the *p*-version in the and -norms and also derive rates of convergence in slightly stronger norms when the exact nature of the singularity of the solution is known. Our results lead to a boundary element method for two-dimensional screen problems in acoustics, which has twice the rate of convergence of the usual *h*-version with uniform mesh. An application to three-dimensional exterior problems is also analyzed.

**[1]**E. Alarcon, L. Abia & A. Reverter, "On the possibility of adaptive boundary elements," in*Accuracy Estimates and Adaptive Refinements in Finite Element Computations*(*AFREC*), Lisbon, 1984.**[2]**E. Alarcon, A. Reverter & J. Molina, "Hierarchical boundary elements,"*Comput. & Structures*, v. 20, 1985, pp. 151-156.**[3]**E. Alarcon & A. Reverter, "*p*-adaptive boundary elements,"*Internat. J. Numer. Methods Engrg.*, v. 23, 1986, pp. 801-829.**[4]**I, Babuška,*The p and h-p Versions of the Finite Element Method. The State of the Art*, Technical Note BN-1156, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 1986.**[5]**I. Babuška, B. A. Szabo, and I. N. Katz,*The 𝑝-version of the finite element method*, SIAM J. Numer. Anal.**18**(1981), no. 3, 515–545. MR**615529**, https://doi.org/10.1137/0718033**[6]**I. Babuška & M. Suri,*The Optimal Convergence Rate of the p-Version of the Finite Element Method*, Technical Note BN-1045, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 1985.**[7]**I. Babuška & M. Suri,*The Treatment of Nonhomogeneous Dirichlet Boundary Conditions by the p-Version of the Finite Element Method*, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 1987.**[8]**I. Babuška and Manil Suri,*The ℎ-𝑝 version of the finite element method with quasi-uniform meshes*, RAIRO Modél. Math. Anal. Numér.**21**(1987), no. 2, 199–238 (English, with French summary). MR**896241**, https://doi.org/10.1051/m2an/1987210201991**[9]**I. Babuška & M. Suri, "The*p*-version of the finite element method for constraint boundary conditions," Institute for Physical Science and Technology, University of Maryland, College Park, MD, 1987.**[10]**C. A. Brebbia, Editor,*Progress in Boundary Element Methods*, Vols. 1, 2, 3, 4, 5, Springer-Verlag, Berlin, 1981 ff.**[11]**Martin Costabel and Ernst Stephan,*Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation*, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR**874845****[12]**Milo R. Dorr,*The approximation theory for the 𝑝-version of the finite element method*, SIAM J. Numer. Anal.**21**(1984), no. 6, 1180–1207. MR**765514**, https://doi.org/10.1137/0721073**[13]**W. Gui and I. Babuška,*The ℎ,𝑝 and ℎ-𝑝 versions of the finite element method in 1 dimension. I. The error analysis of the 𝑝-version*, Numer. Math.**49**(1986), no. 6, 577–612. MR**861522**, https://doi.org/10.1007/BF01389733**[14]**B. Guo & I. Babuška, "The*h-p*version of the finite element method. Part 1: The basic approximation results,"*Comput. Mech.*, v. 1, 1986, pp. 21-41; "Part 2: General results and applications,"*Comput. Mech.*, v. 1, 1986, pp. 203-220.**[15]**Stefan Hildebrandt and Ernst Wienholtz,*Constructive proofs of representation theorems in separable Hilbert space*, Comm. Pure Appl. Math.**17**(1964), 369–373. MR**0166608**, https://doi.org/10.1002/cpa.3160170309**[16]**George C. Hsiao and Wolfgang L. Wendland,*A finite element method for some integral equations of the first kind*, J. Math. Anal. Appl.**58**(1977), no. 3, 449–481. MR**0461963**, https://doi.org/10.1016/0022-247X(77)90186-X**[17]**George C. Hsiao, Ernst P. Stephan, and Wolfgang L. Wendland,*An integral equation formulation for a boundary value problem of elasticity in the domain exterior to an arc*, Singularities and constructive methods for their treatment (Oberwolfach, 1983) Lecture Notes in Math., vol. 1121, Springer, Berlin, 1985, pp. 153–165. MR**806391**, https://doi.org/10.1007/BFb0076269**[18]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Value Problems and Applications*. I, Springer-Verlag, Berlin and New York, 1972.**[19]**Bent E. Petersen,*Introduction to the Fourier transform & pseudodifferential operators*, Monographs and Studies in Mathematics, vol. 19, Pitman (Advanced Publishing Program), Boston, MA, 1983. MR**721328****[20]**E. P. Stephan,*Boundary Integral Equations for Mixed Boundary Value Problems, Screen and Transmission Problems in*, Habilitationsschrift, Technische Hochschule Darmstadt, 1984.**[21]**E. P. Stephan & W. L. Wendland, "Remarks to Galerkin and least squares methods with finite elements for general elliptic problems,"*Manuscripta Geodaetica*, v. 1, 1976, pp. 93-123.**[22]**Ernst P. Stephan and Wolfgang L. Wendland,*An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems*, Applicable Anal.**18**(1984), no. 3, 183–219. MR**767500**, https://doi.org/10.1080/00036818408839520**[23]**E. P. Stephan and W. L. Wendland,*The boundary integral method for two-dimensional screen and crack problems*, Boundary elements, VI (1984), Comput. Mech. Centre, Southampton, 1984, pp. 9.3–18. MR**806607****[24]**W. L. Wendland,*Boundary element methods and their asymptotic convergence*, Theoretical acoustics and numerical techniques, CISM Courses and Lect., vol. 277, Springer, Vienna, 1983, pp. 135–216. MR**762829****[25]**W. L. Wendland,*On some mathematical aspects of boundary element methods for elliptic problems*, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 193–227. MR**811035****[26]**W. L. Wendland,*Splines versus trigonometric polynomials—the ℎ- versus the 𝑝-version in two-dimensional boundary integral methods*, Numerical analysis (Dundee, 1985) Pitman Res. Notes Math. Ser., vol. 140, Longman Sci. Tech., Harlow, 1986, pp. 238–255. MR**873113**

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DOI:
https://doi.org/10.1090/S0025-5718-1989-0947469-5

Article copyright:
© Copyright 1989
American Mathematical Society