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Problem statement

To solve is the following Poisson boundary value problem

$$\begin{cases}\Delta u = f & \mbox{ in } \Omega\subset\mathbb{... ...n } \partial\Omega_N \mbox{ (Neumann boundary)} \end{cases}$$ (1)

In the actual version of the code the following parameters are set:

• $f(x,y) = x^2 + y$ (set in function ASSEMBLE/set_rhs.m)

  Attention: if you are changing the form of the right hand side function $f$ (stored in variable F), then you have to change manually also the jump in the right hand side, stored in variable jf.

• $\Omega$ can have two shapes: `circle' or `flower'.
  • `Circle' is given by
    $$(x-x_c)^2 + (y-y_c)^2 \leq R^2$$
    with some parameters $x_c$, $y_c$ and $R$.
  • The boundary of the domain `flower' is given by
    $$\begin{cases} x & = 0.55(R+0.15\sin(5\theta))\cos\theta + x_c ... ...15\sin(5\theta))\sin\theta + y_c \end{cases} , \theta \in [0, 2\pi) .$$

  Parameters $x_c$, $y_c$ and $R$ are given in ASSEMBLE/problem_setup.m.
• $\partial\Omega$ is splitted into two distinct parts:
  

  $$\partial\Omega_D := \{(x,y)\in\partial\Omega \big{\vert} x<x_{level} \}$$

  $$\partial\Omega_N := \partial\Omega\backslash \partial\Omega_D$$

  
  Parameter $x_{level}$ is set in ASSEMBLE/problem_setup.m.

  Points, where the boundary of $\Omega$ cuts the grid lines are called intersection points. In code they are always stored in structure variables IX. IX_D contains intersection points on $\partial\Omega_D$ and IX_N - the intersection points on $\partial\Omega_N$. Separation of these two sets of intersection points is done in ASSEMBLE/separate_boundaries.m.

• $u_D = x^2$ and $u_N = n_1 + n_2$. The Dirichlet boundary condition function $u_D$ is stored in variable uD, the Neumann boundary condition is stored in uN. (Function ASSEMBLE/set_rhs.m)