A Galerkin Method with Finite Elements for Degenerate One Dimensional Pseudodifferential Equations

Elschner, Johannes

doi:10.1002/mana.19831110104

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…We want to mention that the Galerkin method can converge for degenerate and, in particular, for not strongly elliptic operators if the subprincipal symbol a_1 (x, ) = a2j (x, ) -(2i) a2a2 /9x 0 satisfies additional requirements. The convergence of Galerkin's method with splines for such operators was considered by ELSCHNER [5] and' as a special case of the obtained results one can formulate:…”

Section: Convergence Of the Standard Galerkin Methods Imentioning

confidence: 99%

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The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferential Equations on Closed Curves

Schmidt¹

1984

Z. Anal. Anwend.

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Zcitschrift für Analysis und ihre Anwendungen Bd. 3(4)1984, S. 371-384 The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferent.iai Equations on Closed Curves G. SCHMIDT-'5in der vorliegenden Arbeit wird die näherungsweise Losung von Pseudodifferentialgleichungen nuf geschlossenen Kurven mittels aalerkin-und Kol!okationsverfaliren unters,tcht, die als Ansatzfunktionen polynomiale Splines benut5zen. Es werden hinreichende und im ailgemeinen notwendige Bedingungen für die Konvergenz clieser Verfahren in Sobolevriumen angegeben. I) npeLJ1araeMofi pa6oTe paccMaTpliBaeTcfl npllOJlillueIiIloe pewelilie liceBao,1u 44 epeHl.u r-a2lhuiblx ypaiiiieuitfi iia 3MilTbIX lillBhIX MeToaMII Fa1epI-nIHa It los1jIoi-ca!lIll, B HOTOI)hlX npit6iiuieiiiioe peweulie Illuerca B miac no.1iilloMI4aJ1hHoro cnjiaüiia. JaioTca gocTaToqubie ii, Haii rlpanihrlo, Tai0ie I1C06X0Ll11Mhle yCJJOBIIJI CXO)LIIMOCTI1 3TI1X MCT0OB B HP0CTOIICTB3X Co6oena. The present paper studies the approximate solution of pseudodifferential equations on closed curves using Galerkin and nodal collocatkn methods with polynomial splines; We give sufficient and in general necessary conditions for the convergence of these methods in Sobolcv spaces.

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Section: Convergence Of the Standard Galerkin Methods Imentioning

confidence: 99%

“…Consequently, the set of the right-hand sides, for which Galerkin's method converges, and the order of convergence is smaller than in the strongly elliptic case and this cannot be improved in general (cf. [5]).…”

mentioning

confidence: 99%

The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferential Equations on Closed Curves

Schmidt¹

1984

Z. Anal. Anwend.

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“…(0.1) one uses spline approximations for the unknown function x on F. The two most popular discretization schemes are the Galerkin and collocation methods. If F is a closed curve and c and d are continuous functions, then convergence and error estimates for Galerkin and collocation methods using smooth polynomial splines follow from recent studies by Arnold, Saranen, Stephan and Wendland in [4,23,27], by Nedelec in [14] and by Elschner, Schmidt and the authors in [8,16,17,19,21,22,24,25,26] (see also the surveys just given by Elschner, Pr6gdorf [18] and by Wendland [29]). Convergence results on the spline collocation method in the space L 2 for Eq.…”

Section: C(t) X(t)+d(t)rcz" ! Z-t Rmentioning

confidence: 97%

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

Pröβdorf¹,

Rathsfeld²

1986

Numer. Math.

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Summary. This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients in L 2 on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuniform partitions.

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“…The assertions (i) for s = (1 -1)/2 and (ii) for r 2 (2 -1)/2 are direct consequences of Theorem 6.4.1 and Corollary 6.4.4 in [7] (cf. also [5] in the case of more general finite element spaces). Combining this with the method of proof of Theorem 2.1 in [lS], one easily obtains Theorem 2.1 in the remaining cases.…”

mentioning

confidence: 97%

“…also [5] in the case of more general finite element spaces). Combining this with the method of proof of Theorem 2.1 in [lS], one easily obtains Theorem 2.1 in the remaining cases.…”

mentioning

confidence: 99%

A Galerkin Method with Finite Elements for Degenerate One Dimensional Pseudodifferential Equations

Cited by 3 publications

References 11 publications

The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferential Equations on Closed Curves

The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferential Equations on Closed Curves

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

On Suboptimal Convergence of Finite Element Methods for Pseudodifferential Equations on a Closed Curve

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