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On the theory and error estimation of the reduced basis method for multi- parameter problems. (English) Zbl 0802.65068
The computational analysis of many problems in science and engineering involves the solutions of very large systems of nonlinear equations. Thus, there is considerable interest in reducing the size of these problems. One of these reduction techniques has become generally known as the reduced basis method. The aim of the reduced basis method is to construct special approximations that reduce the number or degrees of freedom without compromising error.
The idea of this method considered by the author may be reviewed as the design of approximations that do account of information about the manifold $$M$$, which is the solution set of a nonlinear equation with multi-parameter, and its local coordinate systems. Therefore, the author thinks that a few basis vectors may already suffice to represent the manifold locally to a relatively high accuracy.
Moreover, the author also proves that with a special projection and a special linear map the approximated problem has a unique solution and the error estimate satisfies an inequality. Finally, the author gives a numerical example of the reduced basis method.

##### MSC:
 65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
PITCON
Full Text:
##### References:
 [1] Nagy, D.A., Model representation of geometrically nonlinear behavior by the finite element method, Comput. struct., 10, 683-688, (1977) · Zbl 0406.73071 [2] Almroth, B.O.; Stern, P.; Brogan, F., Automatic choice of global shape functions in structural analysis, Aiaa j., 16, 525-528, (1978) [3] Noor, A.K.; Peters, J.M., Reduced basis technique for nonlinear analysis of structures, Aiaa j., 18, 455-462, (1981) [4] Noor, A.K., Recent advances in reduction methods for nonlinear problems, Comput. struct., 13, 31-44, (1981) · Zbl 0455.73080 [5] Noor, A.K., On making large nonlinear problems small, Computat. meth. appl. mech. engng, 34, 955-985, (1982) · Zbl 0478.65031 [6] Noor, A.K.; Peters, J.M., Recent advances in reduction methods for instability analysis of structures, Comput. struct., 16, 67-80, (1983) · Zbl 0498.73094 [7] Porsching, T.A.; Lin, Lee M.-Y., The reduced basis method for initial value problems, SIAM J. numer. analysis, 24, 1277-1287, (1987) · Zbl 0639.65039 [8] Lin, Lee M.-Y., Estimation of the error in the reduced basis method solution of differential algebraic equations, SIAM J. numer. analysis, 28, 512-528, (1991) · Zbl 0737.65058 [9] Jarausch, H., Numerical investigation of parabolic differential equations using an adaptive spectral decomposition, (), (In German.) · Zbl 0811.65074 [10] Jarausch, H.; Mackens, W., Solving large nonlinear systems of equations by an adaptive condensation process, Num. math., 50, 633-653, (1987) · Zbl 0647.65036 [11] Mackens, W., Condensation of large sets of nonlinear equations using the reduced basis method, (), (In German.) · Zbl 0551.65059 [12] Noor, A.K.; Peters, J.M., Multiple-parameter reduced basis technique for bifurcation and post-buckling analysis of composite plates, Int. J. numer. meth. engng, 19, 1783-1803, (1983) · Zbl 0557.73070 [13] Noor, A.K.; Balch, C.D.; Shibut, M.A., Reduction methods for nonlinear steady-state thermal analysis, Int. J. numer. meth. engng, 20, 1323-1348, (1984) · Zbl 0557.65076 [14] Noor, A.K.; Peters, J.M.; Andersen, C.M., Mixed models and reduction techniques for large-rotation nonlinear problems, Computat. meth. appl. mech. engng, 44, 67-89, (1984) · Zbl 0517.73073 [15] Noor, A.K.; Andersen, C.M.; Tanner, J.A., Exploiting symmetries in the modeling and analysis of tries, Computat. meth. appl. engng, 63, 37-81, (1987) · Zbl 0634.73077 [16] Fink, J.P.; Rheinboldt, W.C., On the error behavior of the reduced basis technique for nonlinear finite element approximations, Zamm, 63, 21-29, (1983) · Zbl 0533.73071 [17] Porsching, T.A., Estimation of the error in the reduced basis method solution of nonlinear equations, Math. computat., 45, 487-496, (1985) · Zbl 0586.65040 [18] Fink, J.P.; Rheinboldt, W.C., Local error estimates for parametrized nonlinear equations, SIAM J. numer. analysis, 22, 729-735, (1985) · Zbl 0583.65038 [19] Rheinboldt, W.C., Numerical analysis of parametrized equations, () · Zbl 0155.46701 [20] Brezzi, F.; Rappaz, J.; Raviart, P.A., Finite-dimensional approximations of nonlinear problems part I: branches of nonsingular solutions, Num. math., 36, 1-25, (1981) · Zbl 0488.65021 [21] Brezzi, F.; Rappaz, J.; Raviart, P.A., Finite-dimensional approximations of nonlinear problems part II: limit points, Num. math., 37, 1-28, (1981) · Zbl 0525.65036 [22] Brezzi, F.; Rappaz, J.; Raviart, P.A., Finite-dimensional approximations of nonlinear problems part III: simple bifurcation points, Num. math., 38, 1-30, (1981) · Zbl 0525.65037 [23] Rabier, P.J.; Rheinboldt, W.C., On a computational method for the second fundamental tensor and its application to bifurcation problems, Num. math., 57, 681-694, (1990) · Zbl 0699.65037 [24] Mackens, W., Numerical differentiation of implicitly defined space curves, Computing, 41, 237-260, (1987) · Zbl 0666.65016 [25] Rheinboldt, W.C., On the computation of multi-dimensional solution manifolds of parametrized equations, Num. math., 53, 165-181, (1988) · Zbl 0676.65047
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