I am trying to solve a 2D elliptic PDE (see complete electrode model for electrical impedance tomography) using the finite element method (FEM) over a circular region $\Omega$. I have discretized the model, and got the following surface integral

$$\int_e \phi_i(x,y) \phi_j(x,y) \mathrm{d}s, \mbox{ for } i,j = 1,...,n\, .$$

Here $n$ is the number of mesh points, $\phi_i$ are the basis functions and $e$ is an electrode placed on the boundary connecting two adjacent boundary points.

Can I use polar coordinates combined with the standard basis functions in FEM to evaluate these integrals? If so, how can I do this?

  • $\begingroup$ This surface integral is normally calculated by integrating over each element on the face and then summing the results. $\endgroup$ Mar 5 '19 at 11:40
  • $\begingroup$ So, I need to find the element that has the electrode as it's edge, then take the integral over that element. Then I will have an $n \times n$ matrix with a $3\times 3$ submatrix corresponding to the element, for each elements. Am I correct? $\endgroup$ Mar 5 '19 at 17:05
  • $\begingroup$ Sorry, I misread your question! To perform the integral you show above, just select the element with the edge that contains the electrode. Then integrate along that line. The shape functions for the edge can be obtained easily from those of the 2D element. $\endgroup$ Mar 6 '19 at 11:00
  • $\begingroup$ Here is what I have done so far. I got the element that contains the element as an edge. I assigned index numbers to each of the vertices of the element. Suppose the vertices on the electrode are 1 and 9, the other vertex has index 13. So if the integral is taken over this electrode only, and we arrange the matrix, (suppose 20 $\times $20), then only the entries corresponding to these indices will be non-zero, i.e., (1,1), (1,9), (1,13), (9,1), (9,9),...,(13,13) positions will be the only non-zero entries. Please correct me if I am wrong. $\endgroup$ Mar 7 '19 at 1:21
  • $\begingroup$ I am writing a code for the forward problem, and I need help in clarifying this issue. $\endgroup$ Mar 7 '19 at 18:57

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