I am using MATLAB's PDE toolbox to solve the differential equation$-\nabla\cdot\left(c(x)\nabla u(x)\right) + a(x)u(x) = f(x)$The particular problem in question is an electrostatic problem, but has the solution (voltage) specified at several points inside the domain. Since the PDE tool is setup to apply boundary conditions along edges, I thought I might apply these point conditions by the following process:For a boundary condition at point $x_0$, where we want to specify $u(x_0) = b_0$,$c(x_0) = 0$$a(x_0) = 1$$f(x_0) = b_0$Such that at the point $x = x_0$, the above differential equation simply becomes $u(x) = b_0$.However, this does not appear to result in the correct solution. Is this a mathematically valid process? If not, is there a preferred method of setting the solution values at certain arbitrary points in the PDE tool? I am not using the GUI.EditIn my understanding, this is a fairly straightforward process using the FEM method. One merely specifies that the corresponding row of the stiffness matrix [K] is entirely 0, with the exception of a 1 at the index of the corresponding node. Then the value of that point is placed in the corresponding row of the solution [b] matrix. Unfortunately, these matrices are not accessible in the PDE tool, hence the question of how to proceed.ResolutionThank you all for your help. In the end, it appears that prescribed value points are not currently compatible with MATLAB's adaptmesh algorithm. I ended up writing my own mesh adaption algorithm and using the assempde commands to specify the matrix data and solve.
Prescribe solution of a PDE at specific points
matlab;finite element;constraints