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As far as I know, for getting a unique solution to a PDE we should impose some boundary conditions to the PDE. "The number of required auxiliary conditions is determined by the highest order derivative in each independent variable. "

My questions are: In any numerical method, especially in finite element method, is it valid:

1-) to use more boundary conditions than the highest order derivative in each independent variable.

2-) to use less boundary conditions than the highest order derivative in each independent variable.

For example:

1-) In this article and in this article, while the highest order derivative of a spatial variable is in third order authors used two boundary conditions for that variable.

2-) In this article while the highest order derivative of a spatial variable is in third order authors used four boundary conditions for that variable.

I can increase the number of articles like this.

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  • $\begingroup$ You should take into account that some (homogeneous) boundary conditions are satisfied by the weak form of the PDE. These are termed natural boundary conditions. $\endgroup$ – nicoguaro Aug 29 '15 at 20:09
  • $\begingroup$ I am aware of that but I think the issue is not relevant with natural bc's. Also what about the numerical methods that do not use weak form such as collocation methods? $\endgroup$ – Ömer Aug 29 '15 at 20:31
  • $\begingroup$ They can be written in weak form. Select a Dirac delta as weight in your weighted-residuals form. $\endgroup$ – nicoguaro Aug 29 '15 at 20:34
  • $\begingroup$ They may be written in a weak form ok. but also they widely used directly and I see in some papers that the number of boundary conditions do not match with the order of pde. On the other hand as far as I know, all of the boundary conditions (natural or essential) must be specified at the time that the PDE problem is specified. $\endgroup$ – Ömer Aug 29 '15 at 21:47
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    $\begingroup$ I'm afraid in this generality, the only answer to your question can be "as many as necessary to have a unique solution". Note that the articles you link to are about nonlinear PDEs, for which there's no single theory. $\endgroup$ – Christian Clason Aug 30 '15 at 9:03