Projection methods with different trial and test spaces
Author:
M. S. Mock
Journal:
Math. Comp. 30 (1976), 400416
MSC:
Primary 65N30
MathSciNet review:
0423840
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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, 1967. MR 0239327
(39 #684)
 [2]
A.
K. Aziz (ed.), The mathematical foundations of the finite element
method with applications to partial differential equations, Academic
Press, New York, 1972. MR 0347104
(49 #11824)
 [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 WisconsinMadison,
Academic Press, New York, 1974, pp. 1–14. Publication No. 33. MR 0657964
(58 #31903)
 [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
(51 #9528)
 [6]
Jim
Douglas Jr. and Todd
Dupont, Galerkin methods for parabolic equations, SIAM J.
Numer. Anal. 7 (1970), 575–626. MR 0277126
(43 #2863)
 [7]
Jim
Douglas Jr. and Todd
Dupont, A finite element collocation method
for quasilinear parabolic equations, Math.
Comp. 27 (1973),
17–28. MR
0339508 (49 #4266), http://dx.doi.org/10.1090/S00255718197303395088
 [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
(48 #10130)
 [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
(50 #15360)
 [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. R2, 61–66 (English, with Loose
French summary). MR 0359358
(50 #11812)
 [12]
Carl
de Boor (ed.), Mathematical aspects of finite elements in partial
differential equations, Academic Press [A subsidiary of Harcourt Brace
Jovanovich, Publishers], New YorkLondon, 1974. Publication No. 33 of the
Mathematics Research Center, The University of WisconsinMadison. MR 0349031
(50 #1525)
 [13]
Todd
Dupont, Galerkin methods for first order hyperbolics: an
example, SIAM J. Numer. Anal. 10 (1973),
890–899. MR 0349046
(50 #1540)
 [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
(53 #7067)
 [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
(28 #2414)
 [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., WoltersNoordhoff, Groningen, 1972. MR 41 #4271.
 [17]
P.
Lesaint, Finite element methods for symmetric hyperbolic
equations, Numer. Math. 21 (1973/74), 244–255.
MR
0341902 (49 #6648)
 [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
(51 #4683)
 [19]
M.
S. Mock, A global a posteriori error estimate for quasilinear
elliptic problems, Numer. Math. 24 (1975),
53–61. MR
0471364 (57 #11098)
 [20]
M.
S. Mock, Explicit finite element schemes for first order symmetric
hyperbolic systems, Numer. Math. 26 (1976),
no. 4, 367–378. MR 0448955
(56 #7260)
 [21]
J.
Nitsche, Ein Kriterium für die QuasiOptimalität des
Ritzschen Verfahrens, Numer. Math. 11 (1968),
346–348 (German). MR 0233502
(38 #1823)
 [22]
Carl
de Boor (ed.), Mathematical aspects of finite elements in partial
differential equations, Academic Press [A subsidiary of Harcourt Brace
Jovanovich, Publishers], New YorkLondon, 1974. Publication No. 33 of the
Mathematics Research Center, The University of WisconsinMadison. MR 0349031
(50 #1525)
 [23]
R.
D. Russell and L.
F. Shampine, A collocation method for boundary value problems,
Numer. Math. 19 (1972), 1–28. MR 0305607
(46 #4737)
 [24]
Martin
H. Schultz, 𝐿² error bounds for the
RayleighRitzGalerkin method, SIAM J. Numer. Anal. 8
(1971), 737–748. MR 0298918
(45 #7967)
 [25]
Gilbert
Strang and George
J. Fix, An analysis of the finite element method,
PrenticeHall Inc., Englewood Cliffs, N. J., 1973. PrenticeHall Series in
Automatic Computation. MR 0443377
(56 #1747)
 [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
(53 #7077)
 [27]
Vidar
Thomée and Lars
Wahlbin, On Galerkin methods in semilinear parabolic problems,
SIAM J. Numer. Anal. 12 (1975), 378–389. MR 0395269
(52 #16066)
 [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
(51 #7309)
 [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
(37 #1762)
 [1]
 J. H. AHLBERG, E. N. NILSON & J. L. WALSH, The Theory of Splines and Their Applications, Academic Press, New York, 1967. MR 39 #684. MR 0239327 (39:684)
 [2]
 I. BABUšKA & A. K. AZIZ, "Survey lectures on the mathematical foundations of the finite element method," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (editor), Academic Press, New York, 1972. MR 0347104 (49:11824)
 [3]
 J. H. BRAMBLE & A. M. SCHATZ, "Higher order local accuracy by averaging in the finite element method," Mathematical Aspects of Finite Elements in Partial Differential Equation, C. deBoor (editor), Academic Press, New York, 1974. MR 0657964 (58:31903)
 [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]
 C. deBOOR & B. SWARTZ, "Collocation at Gaussian points," SIAM J. Numer. Anal., v. 10, 1973, pp. 582606. MR 0373328 (51:9528)
 [6]
 J. DOUGLAS, JR. & T. DUPONT, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 43 #2863. MR 0277126 (43:2863)
 [7]
 J. DOUGLAS, JR. & T. DUPONT, "A finite element collocation method for quasilinear parabolic equations," Math. Comp., v. 27, 1973, pp. 1728. MR 0339508 (49:4266)
 [8]
 J. DOUGLAS, JR. & T. DUPONT, "Superconvergence for Galerkin methods for the two point boundary problem via local projections," Numer. Math., v. 21, 1973, pp. 270278. MR 0331798 (48:10130)
 [9]
 J. DOUGLAS, JR. & T. DUPONT, "Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces," Numer. Math., v. 22, 1974, pp. 99109. MR 0362922 (50:15360)
 [10]
 J. DOUGLAS, JR., T. DUPONT, H. H. RACHFORD, JR. & M. F. WHEELER, " Galerkin methods for problems involving several space variables." (To appear.)
 [11]
 J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, "An estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials," Rev. Francaise Automatique Informatique et Recherche Operationnelle, v. R2, 1974, pp. 6166. MR 0359358 (50:11812)
 [12]
 J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, "Galerkin methods for the Laplace and heat equations," Mathematical Aspects of Finite Elements in Partial Differential Equations, C. deBoor (editor), Academic Press, New York, 1974. MR 0349031 (50:1525)
 [13]
 T. DUPONT, "Galerkin methods for first order hyperbolics: An example," SIAM J. Numer. Anal., v. 10, 1973, pp. 890899. MR 50 #1540. MR 0349046 (50:1540)
 [14]
 T. DUPONT, "Some error estimates for parabolic Galerkin methods," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (editor), Academic Press, New York, 1972, pp. 491504. MR 0403255 (53:7067)
 [15]
 M. A. KRASNOSEL'SKIĬ, Topological Methods in the Theory of Nonlinear Integral Equations, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR 20 #3464; 28 #2414. MR 0159197 (28:2414)
 [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., WoltersNoordhoff, Groningen, 1972. MR 41 #4271.
 [17]
 P. LESAINT, "Finite element methods for symmetric hyperbolic equations," Numer. Math., v. 21, 1973/74, pp. 244255. MR 49 #6648. MR 0341902 (49:6648)
 [18]
 T. R. LUCAS & G. W. REDDIEN, "A high order projection method for nonlinear two point boundary value problems," Numer. Math., v. 20, 1973, pp. 257270. MR 0368442 (51:4683)
 [19]
 M. S. MOCK, "A global a posteriori error estimate for quasilinear elliptic problems," Numer. Math., v. 24, 1975, pp. 5361. MR 0471364 (57:11098)
 [20]
 M. S. MOCK, "Explicit finite element schemes for first order symmetric hyperbolic systems," Numer. Math. (To appear.) MR 0448955 (56:7260)
 [21]
 J. NITSCHE, "Ein Kriterium für die QuasiOptimalitä't des Ritzschen Verfahrens," Numer. Math., v. 11, 1968, pp. 346348. MR 38 #1823. MR 0233502 (38:1823)
 [22]
 H. H. RACHFORD, JR. & M. F. WHEELER, "An Galerkin procedure for the twopoint boundary value problem," Mathematical Aspects of Finite Elements in Partial Differential Equations, C. deBoor (editor), Academic Press, New York, 1974. MR 0349031 (50:1525)
 [23]
 R. D. RUSSELL & L. F. SHAMPINE, "A collocation method for boundary value problems," Numer. Math., v. 19, 1972, pp. 128. MR 46 #4737. MR 0305607 (46:4737)
 [24]
 M. H. SCHULTZ, " error bounds for the RaleighRitzGalerkin method," SIAM J. Numer. Anal., v. 8, 1971, pp. 737748. MR 45 #7967. MR 0298918 (45:7967)
 [25]
 G. STRANG & G. J. FIX, An Analysis of the Finite Element Method, PrenticeHall, Englewood Cliffs, N.J., 1973. MR 0443377 (56:1747)
 [26]
 V. THOMÉE, "Spline approximation and difference schemes for the heat equation," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. K. Aziz (editor), Academic Press, New York, 1972, pp. 711746. MR 0403265 (53:7077)
 [27]
 V. THOMÉE & L. WAHLBIN, "On Galerkin methods in semilinear parabolic problems," SIAM J. Numer. Anal., v. 12, 1975, pp. 378389. MR 0395269 (52:16066)
 [28]
 V. THOMÉE & B. WENDROFF, "Convergence estimates for Galerkin methods for variable coefficient initial value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 10591068. MR 0371088 (51:7309)
 [29]
 M. YAMAGUTI & T. NOGI, "An algebra of pseudo difference schemes and its application," Publ. Res. Inst. Math. Sci. Ser. A, v. 3, 1967/68, pp. 151166. MR 37 #1762. MR 0226172 (37:1762)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N30
Retrieve articles in all journals
with MSC:
65N30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197604238406
PII:
S 00255718(1976)04238406
Article copyright:
© Copyright 1976 American Mathematical Society
