Abstract
The Ritz method for strong elliptic pseudo-differential equations is discussed. ‘Optimal’ local error estimates are derived if the underlying ‘approximation-spaces’ are finite elements. The analysis covers simultaneously pseudo-differential operators of positive and negative order. In case of positive order an additional regularity assumption for the ‘approximation-spaces’ is needed.
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References
G. Alexits, Einige Beiträge zur Approximationstheorie. Acta Sci. Math.,16 (1965), 212–224.
J. P. Aubin, Approximation des espaces de distributions et des opérateurs différentiels. Bull. Soc. Math. France, Suppl.12 (1967).
A. K. Aziz and R. B. Kellog, Finite element analysis of scattering problem. Math. Comp.,37 (1981), 261–272.
I. Babuska, Error bounds for finite element method. Numer. Math.,16 (1971), 322–333.
J. H. Bramble and R. Scott, Simultaneous approximation in scales of Banach spaces. Math. Comp.,32 (1978), 947–954.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg-New York, 1977.
H. P. Helfrich, Simultaneous approximation in negative norms of arbitrary order. RAIRO Anal. Numér.,15 (1981), 231–235.
G. C. Hsiao and W. L. Wendland, A finite element method for some integral equations of the first kind. J. Math. Anal. Appl.,58 (1977), 449–481.
J.J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators. Comm. Pure Appl. Math.,18 (1965), 269–305.
S. G. Krein and I. Petunin, Scales of Banach spaces. Russian Math. Surveys,21 (1966), 85–159.
J. Nitsche, Ein Kriterium für die Quasi-Optimalität des Ritz’schen Verfahrens. Numer. Math.,11 (1968), 346–348.
J. Nitsche, Zur Konvergenz von Näherungsverfahren bezüglich verschiedener Normen. Numer. Math.,15 (1970), 224–228.
J. Nitsche and A. H. Schatz, On local approximation properties ofL 2-projections on spline subspaces. Applicable Anal.,2 (1972), 161–168.
J. Nitsche and A. H. Schatz, Interior estimates for Ritz-Galerkin methods. Math. Comp.,28 (1974), 937–958.
N. I. Polskii, On a general scheme of application of approximation methods. Dokl. Akad. Nauk,111 (1956), 1181–1184.
A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp.,28 (1974), 959–962.
E. Stephan and W. L. Wendland, Remarks to Galerkin and least squares methods with finite elements for general elliptic problems. Manuscripta Geodaetica,1 (1976), 93–123.
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators I, II. Plenum Press, New York-London, 1980.
G. Vainikko, On the question of convergence of Galerkin’s method. Tartu Riikl. Ül. Toimetised,177 (1965), 148–152.
M. Zlamal, Curved elements in the finite element method. Part I: SIAM J. Numer. Anal.,10 (1973), 229–240, Part II: SIAM J. Numer. Anal.,11 (1974), 347–362.
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Braun, K. Interior error estimates of the Ritz method for pseudo-differential equations. Japan J. Appl. Math. 3, 59–72 (1986). https://doi.org/10.1007/BF03167092
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DOI: https://doi.org/10.1007/BF03167092