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What are the basic difficulties of using the collocation techniques for solving PDE’s?

B.Banerjee's picture

Hello, Can anybody inform me what are the basic difficulties of using point collocation (strong form) kind of method for solving pde's when compared with solving its weak statement? I have listed a few, known to me,

1. Required higher order continuity.

2. Approximation (field variables) may have spurious boundary oscillations (so called Gibbs phenomenon)

3. Applying Neumann boundary condition is not so trivial.


B. Banerjee


N. Sukumar's picture

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 K is no longer strictly positive definite and spectral convergence is lost. Here's a recent paper that talks about RBFs and the Gibbs phenomena.

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