Hello Ehsan,
I'm a PhD student in Germany and I try to use your hp-VPINN approach for 2D convection-dominated convection-diffusion equations. I've implemented your idea by myself based on this code and what you wrote in your paper. I've got a small questions that I couldn't find an answer the last days. If I understand your paper correctly, then you are using as test functions P_{k+1}-P_{k-1}, where P_{k} is the k-th Legendre polynomial, to ensure that they are zero along the boundary of the cell(s). Furthermore, in section 4.1 you write that you are using Gauss-Lobatto integration and this is what I see in your code as well. I was wondering why you use Gauss-Lobatto and not Gauss-Legendre? Using Gauss-Lobatto seems to be strange to me. Gauss-Lobatto includes the boundary of the domain but by construction the test functions are zero there. Why is that important in contrast to Gauss-Legendre?
Thanks a lot in advance and thank you for your research for your work! :)
Best
Derk
Hello Ehsan,
I'm a PhD student in Germany and I try to use your hp-VPINN approach for 2D convection-dominated convection-diffusion equations. I've implemented your idea by myself based on this code and what you wrote in your paper. I've got a small questions that I couldn't find an answer the last days. If I understand your paper correctly, then you are using as test functions P_{k+1}-P_{k-1}, where P_{k} is the k-th Legendre polynomial, to ensure that they are zero along the boundary of the cell(s). Furthermore, in section 4.1 you write that you are using Gauss-Lobatto integration and this is what I see in your code as well. I was wondering why you use Gauss-Lobatto and not Gauss-Legendre? Using Gauss-Lobatto seems to be strange to me. Gauss-Lobatto includes the boundary of the domain but by construction the test functions are zero there. Why is that important in contrast to Gauss-Legendre?
Thanks a lot in advance and thank you for your research for your work! :)
Best
Derk