Projection methods with different trial and test spaces

Author:
M. S. Mock

Journal:
Math. Comp. **30** (1976), 400-416

MSC:
Primary 65N30

MathSciNet review:
0423840

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Abstract: We consider finite element projection methods for linear partial differential equations, in which the spaces of trial functions and test functions may be different. In addition to approximation and smoothness properties, conditions implying equality of dimensions and uniform coerciveness are required, the most important of which resembles a strong form of an inverse assumption.

Our results provide a mechanism for the difference in the rate of convergence of Galerkin procedures with cubic splines and Hermite cubics, applied to first order symmetric hyperbolic problems [13].

**[1]**J. H. Ahlberg, E. N. Nilson, and J. L. Walsh,*The theory of splines and their applications*, Academic Press, New York-London, 1967. MR**0239327****[2]**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****[3]**J. H. Bramble and A. H. Schatz,*Higher order local accuracy by averaging in the finite element method*, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 1–14. Publication No. 33. MR**0657964****[4]**C. deBOOR,*The Method of Projections as Applied to the Numerical Solution of Two Point Boundary Value Problems Using Cubic Splines*, Ph. D. Thesis, University of Michigan, 1966.**[5]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328****[6]**Jim Douglas Jr. and Todd Dupont,*Galerkin methods for parabolic equations*, SIAM J. Numer. Anal.**7**(1970), 575–626. MR**0277126****[7]**Jim Douglas Jr. and Todd Dupont,*A finite element collocation method for quasilinear parabolic equations*, Math. Comp.**27**(1973), 17–28. MR**0339508**, 10.1090/S0025-5718-1973-0339508-8**[8]**Jim Douglas Jr. and Todd Dupont,*Superconvergence for Galerkin methods for the two point boundary problem via local projections*, Numer. Math.**21**(1973/74), 270–278. MR**0331798****[9]**Jim Douglas Jr. and Todd Dupont,*Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces*, Numer. Math.**22**(1974), 99–109. MR**0362922****[10]**J. DOUGLAS, JR., T. DUPONT, H. H. RACHFORD, JR. & M. F. WHEELER, " Galerkin methods for problems involving several space variables." (To appear.)**[11]**Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler,*An 𝐿^{∞} estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér Rouge**8**(1974), no. R-2, 61–66 (English, with Loose French summary). MR**0359358****[12]**Carl de Boor (ed.),*Mathematical aspects of finite elements in partial differential equations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR**0349031****[13]**Todd Dupont,*Galerkin methods for first order hyperbolics: an example*, SIAM J. Numer. Anal.**10**(1973), 890–899. MR**0349046****[14]**Todd Dupont,*Some 𝐿² error estimates for parabolic Galerkin methods*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 491–504. MR**0403255****[15]**M. A. Krasnosel’skii,*Topological methods in the theory of nonlinear integral equations*, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR**0159197****[16]**M. A. KRASNOSEL'SKIĬ, G. M. VAĬNIKKO, P. P. ZABREĬKO, Ja. B. RUTICKIĬ & V. Ja. STECENKO,*Approximate Solution of Operator Equations*, "Nauka", Moscow, 1969; English transl., Wolters-Noordhoff, Groningen, 1972. MR**41**#4271.**[17]**P. Lesaint,*Finite element methods for symmetric hyperbolic equations*, Numer. Math.**21**(1973/74), 244–255. MR**0341902****[18]**Thomas R. Lucas and George W. Reddien,*A high order projection method for nonlinear two point boundary value problems*, Numer. Math.**20**(1972/73), 257–270. MR**0368442****[19]**M. S. Mock,*A global a posteriori error estimate for quasilinear elliptic problems*, Numer. Math.**24**(1975), 53–61. MR**0471364****[20]**M. S. Mock,*Explicit finite element schemes for first order symmetric hyperbolic systems*, Numer. Math.**26**(1976), no. 4, 367–378. MR**0448955****[21]**J. Nitsche,*Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens*, Numer. Math.**11**(1968), 346–348 (German). MR**0233502****[22]**Carl de Boor (ed.),*Mathematical aspects of finite elements in partial differential equations*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR**0349031****[23]**R. D. Russell and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607****[24]**Martin H. Schultz,*𝐿² error bounds for the Rayleigh-Ritz-Galerkin method*, SIAM J. Numer. Anal.**8**(1971), 737–748. MR**0298918****[25]**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377****[26]**Vidar Thomée,*Spline approximation and difference schemes for the heat equation*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 711–746. MR**0403265****[27]**Vidar Thomée and Lars Wahlbin,*On Galerkin methods in semilinear parabolic problems*, SIAM J. Numer. Anal.**12**(1975), 378–389. MR**0395269****[28]**Vidar Thomée and Burton Wendroff,*Convergence estimates for Galerkin methods for variable coefficient initial value problems*, SIAM J. Numer. Anal.**11**(1974), 1059–1068. MR**0371088****[29]**Masaya Yamaguti and Tatsuo Nogi,*An algebra of pseudo difference schemes and its application*, Publ. Res. Inst. Math. Sci. Ser. A**3**(1967/1968), 151–166. MR**0226172**

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0423840-6

Article copyright:
© Copyright 1976
American Mathematical Society