I am trying to solve a second order pde where both the boundary conditions are Dirichlet type and the value of the dependent variable is zero at both the boundary. Can I use boundary collocation method to solve the problem?
Definitely you can use since zero is excepted value unless there is singularity there in some of those extra terms in the differential equation used in the boundary value problem. I am using indirect boundary element method having straight line elements for 2D problems (i.er., the flat triangular for 3D'' in very complicated system, which is subjected to the electrostatic and elastostatic forces which varies in space and time domain, very successfully., last 25 years. In the case of elastic problem one has to be very careful in scaling the computation domain.
To understand the collocation method and it properly, use the weighted residual approach. For example, Brebbia, C. A. and Dominguez, J., 2001, Boundary Elements. An introduction course. You may see also simple explanation in my book Classical and Modern Engineering Methods in Fluid Flow and Heat Transfer,2013, p.254
Dear Abram, I prefer direct application of the Greens function using the discrete representation of the domain of interest including the internal and external boundaries even if one has additional body forces acting in the interior region. That means an extension of the Indirect Boundary Element method using the simple straight elements in 2D manifold and the flat trigonal elements for the 3D system. Especially if one has moving boundary problem such as the case for a vacancy in a matrix under the electromigration forces as well as it is exposed simultaneously to the surface mass flow induced by the local temperature gradient associated with the steady state heat flow. Best Regards. This problem is treated above given publications, and which deal with the failure of interconnects driven by capillary, electromigration, and thermal-stress gradients induced by steady-state heat flow in passivated metallic thin films and flip chip solder joints.
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