iMechanica - Comments for "What are the basic difficulties of using the collocation techniques for solving PDE’s?"
https://imechanica.org/node/1042
Comments for "What are the basic difficulties of using the collocation techniques for solving PDE’s?"enCollocation schemes
https://imechanica.org/comment/1686#comment-1686
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<p><em>In reply to <a href="https://imechanica.org/node/1042">What are the basic difficulties of using the collocation techniques for solving PDE’s?</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Here are a few more that come to mind (some related to those you have mentioned). Difficult to prove convergence of collocation methods in general and to bound the associated errors. The weak/variational form for FEM enables one to bound the error and to ensure convergence to the exact solution (esp. for linear problems ) that is one-sided with refinement (more basis functions via h-refinement or higher-order elements/approximation). In finite-differences too such statements on the errors can not be made (errors can be positive or negative). Collocation schemes using radial basis functions (RBFs) are well-known and they are amenable to mathematical (convergence) analysis (global RBFs). Even here only for a select few global bases (e.g., multiquadrics or Gaussian) is the stiffness matrix invertible (strictly positive-definite) and hence the data approximation problem (i.e., essential boundary conditions to solve a PDE) is solvable. The stiffness matrix becomes fully populated (unlike weak formulations using compactly-supported basis functions as in FEM) in collocation schemes and handling natural BCs also requires extra work and care (unlike weak formulations). A big plus of these is that spectral convergence is obtained (global bases), and hence fewer nodes/unknowns suffice to get very good accuracy. The trade-off when one attempts to use compactly-supported RBFs is that <strong>K</strong> is no longer strictly positive definite and spectral convergence is lost. Here's a <a href="http://portal.acm.org/citation.cfm?id=1224236.1224273&coll=p&dl=GUIDE&CFID=15151515&CFTOKEN=6184618" target="_blank">recent paper</a> that talks about RBFs and the Gibbs phenomena.</p>
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</ul>Wed, 14 Mar 2007 07:25:01 +0000N. Sukumarcomment 1686 at https://imechanica.org