Local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on curves

Author:
Jukka Saranen

Journal:
Math. Comp. **48** (1987), 485-502

MSC:
Primary 65R20; Secondary 35S99, 45J05, 65L10, 65N35

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878686-9

MathSciNet review:
878686

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Abstract: In this article we derive local error estimates for some Petrov-Galerkin methods applied to strongly elliptic equations on smooth curves of the plane. The results, e.g., cover the basic first-kind and second-kind integral equations appearing in the boundary element solution of the potential problem. The discretization model includes the Galerkin method and the collocation method using smoothest splines as trial functions. Asymptotic error estimates are given for a large scale of the Sobolev norms.

**[1]**Douglas N. Arnold and Wolfgang L. Wendland,*On the asymptotic convergence of collocation methods*, Math. Comp.**41**(1983), no. 164, 349–381. MR**717691**, https://doi.org/10.1090/S0025-5718-1983-0717691-6**[2]**Douglas N. Arnold and Wolfgang L. Wendland,*The convergence of spline collocation for strongly elliptic equations on curves*, Numer. Math.**47**(1985), no. 3, 317–341. MR**808553**, https://doi.org/10.1007/BF01389582**[3]**A. K. Aziz (ed.),*The mathematical foundations of the finite element method with applications to partial differential equations*, Academic Press, New York-London, 1972. MR**0347104****[4]**Garrett Birkhoff,*Local spline approximation by moments*, J. Math. Mech.**16**(1967), 987–990. MR**0208241****[5]**Carl de Boor,*On the local spline approximation by moments*, J. Math. Mech.**17**(1967/1968), 729–735. MR**0223803****[6]**J. Descloux,*Interior regularity and local convergence of Galerkin finite element approximations for elliptic equations*, Topics in numerical analysis, II (Proc. Roy. Irish Acad. Conf., Univ. College, Dublin, 1974) Academic Press, London, 1975, pp. 27–41. MR**0413544****[7]**Jim Douglas Jr., Todd Dupont, and Lars Wahlbin,*Optimal 𝐿_{∞} error estimates for Galerkin approximations to solutions of two-point boundary value problems*, Math. Comp.**29**(1975), 475–483. MR**0371077**, https://doi.org/10.1090/S0025-5718-1975-0371077-0**[8]**J. Elschner & G. Schmidt,*On Spline Interpolation in Periodic Sobolev Spaces*, Preprint 01/83, Dept. Math. Akademie der Wissenschaften der DDR.**[9]**G. C. Hsiao and W. L. Wendland,*The Aubin-Nitsche lemma for integral equations*, J. Integral Equations**3**(1981), no. 4, 299–315. MR**634453****[10]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Value Problems and Applications I*, Springer-Verlag, Berlin and New York, 1972.**[11]**J. Nitsche and A. Schatz,*On local approximation properties of 𝐿₂-projection on spline-subspaces*, Applicable Anal.**2**(1972), 161–168. Collection of articles dedicated to Wolfgang Haack on the occasion of his 70th birthday. MR**0397268**, https://doi.org/10.1080/00036817208839035**[12]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**[13]**R. Rannacher & W. L. Wendland,*On the Order of Pointwise Convergence of Some Boundary Element Methods*, Part I, Preprint Nr. 760, Technische Hochschule Darmstadt, Germany, 1983.**[14]**J. Saranen and W. L. Wendland,*On the asymptotic convergence of collocation methods with spline functions of even degree*, Math. Comp.**45**(1985), no. 171, 91–108. MR**790646**, https://doi.org/10.1090/S0025-5718-1985-0790646-3**[15]**R. Seeley,*Topics in pseudo-differential operators*, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) Edizioni Cremonese, Rome, 1969, pp. 167–305. MR**0259335****[16]**François Trèves,*Introduction to pseudodifferential and Fourier integral operators. Vol. 2*, Plenum Press, New York-London, 1980. Fourier integral operators; The University Series in Mathematics. MR**597145**

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0878686-9

Article copyright:
© Copyright 1987
American Mathematical Society