Hello Wolfgang, Thank you for your reply. What you described above is the first approach that I took. There are some differences but overall, it is the same thing. I am not familiar with the form:
div E - Delta V = 0 but, I was able to figure out how you came this. Instead, I kept the equations separate and applied the technique described in lecture 19. For reference, the equations that I am referring to are: -grad E = 0 delta V = 0 where E is the electric field vector. I saw a new approach that I could take which I will describe here. Coming back to the original equations: delta V = 0 -grad V = E And then applying lecture 19 technique: (delta phi_V, V) - (grad phi_V, V) = (E, phi_V) However, I run into the problem which I believe you briefly touched upon on your answer which is I am multiplying by a single test function; phi_V. So I am also thinking that this is the wrong approach. There is a third equation that I have not touched yet which is the curl E = 0. If I were to account for this equation, then this would become the third approach to the problem. Thus the "complete" set of equations would be delta V = 0 -grad V = E curl E = 0 If curl E = 0 is taken into account, then I would be working with the form: (delta phi_V, V) - (grad phi_V, V) + (curl phi_E, E) = (phi_V, E) Now, it would seem that i am working with two test functions, phi_V and phi_E. So it would appear that this form is more appropriate to work with. Is this the direction that I should be taking or is there another approach that I should consider? On Tuesday, September 4, 2018 at 5:25:43 PM UTC-4, Wolfgang Bangerth wrote: > > > > So far, I have found that there are two paths that I can take to get my > > bilinear form. I have already attempted one path and I ran into a > compile > > error with Deal.II. So, I am thinking that this bilinear form may not be > > appropriate. > > That's not necessarily the right conclusion. Just because you chose the > wrong > syntax (= compiler error) does not mean that you chose the logically wrong > approach. > > > > But I wanted to turn to the community here to see if my bilinear > > form has an issue or if my code has an issue. > > > > For the first path, since V and E are my unknowns, I wanted both V and E > to be > > on the left-hand side of their PDEs. With a bit of substitution, the > equations > > that I would be solving turned into this: > > I don't see your inlined equations in my mail program, but assuming that > you > are talking about the equation > > (div E, phi_E) - (grad phi_V, grad V) = 0 > > then this is not correct. If you start out from the equation > div E - Delta V = 0 > then you will want to multiply this (scalar) equation with a test function > and > integrate by parts where necessary. But it's *one* test function, so you > need > to end up multiplying *both* terms with the same test function phi_V, not > different test functions for the two terms. > > That said, here are two questions: > * In essence, you want to solve the Laplace equation for V, but derive a > mixed > form, which with your variables is usually stated as > E + grad V = 0 > whereas you choose > div E + Delta V = 0 > This raises the question of why you want to do it this way? You are > requiring more differentiability than necessary, negating the > advantages of > the mixed form. > > * If you chose to go with the formulation you have, I suspect that you > want to > integrate by parts: > - (E, grad phi_V) - (grad phi_V, grad V) = 0 > so that all derivatives are on the V test functions, not on E. Any good > reasons not to do so? I'm not sure that your formulation is uniquely > solvable at all. > > Best > W. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: [email protected] > <javascript:> > www: http://www.math.colostate.edu/~bangerth/ > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. For more options, visit https://groups.google.com/d/optout.
