Lecture 4: 15 September 2010Lecture 4: 15 September 2010 Compared the 1d Laplacian to its discrete approximation. Both have real, negative eigenvalues, both have orthogonal eigenvectors/eigenfunctions. In the discrete Laplacian, the real eigenvalues and the orthogonal eigenvectors stem from the fact that it is (obviously) real-symmetric; the same fact also implies that it is diagonalizable (complete basis of eigenvectors). The Laplacian is not (yet) obviously real-symmetric, whatever that means for a derivative, but it is "diagonalizable-ish"; its eigenfunctions span anything with a Fourier sine series (i.e. anything that does not diverge too quickly), as long as we ignore finite errors at isolated points of discontinuity. (Pinning down diagonalizability of differential operators is a difficult topic, one that requires half a functional analysis course even for relatively simple cases.) The negative eigenvalues mean that the discrete Laplacian is negative definite, and also suggest that it can be written in the form -DTD for some D. Show that this is indeed the case: we derived the discrete Laplacian by turning two derivatives into differences, one by one, by writing the first step as a matrix we get D, and writing the second step as a matrix shows that it is -DT. Reviewed the proof that this means the matrix is negative definite, which also relies on D being full column rank (which is easy to see from DT since it is upper-triangular). To do a similar analysis of the actual Laplacian, we first have to have a dot product, or inner product. Defined an abstract 〈u,v〉 notation for inner products, as well as three key properties. First, 〈u,v〉 = complex conjugate of 〈v,u〉. Second, |u|2=〈u,u〉 must be nonnegative, and zero only if u=0. Third, it must be linear: 〈u,αv+βw〉=α〈u,v〉+β〈u,w〉 (note: some textbooks, especially in functional analysis, put the conjugation on the second argument instead of the first). For functions, the most common inner product (though not the only choice, as you will see in pset 2) is a simple integral; we will look at this more next time.MIT OpenCourseWarehttp://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and NumericsFall 2010 For information about citing these materials or our Terms of Use, visit:
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