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Numerical Simulation

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M. Griebel

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Boundary Conditions

Consider, the Poisson problem

$$-\Delta u = f \quad\mbox{ in }\quad ]0,1[^d$$ (5.1)

with Dirichlet-, Neumann- or periodic boundary conditions (BC) on each of the domain's faces. Our principal approach is to reduce (5.1) to a problem with homogeneous Dirichlet-, Neumann- or periodic BC. To this end, $u$ is considered a sum $u=u_0 + \bar{u}$, where $\bar{u}$ is a function which takes the eventual inhomogeneous Dirichlet or Neumann BC and $u_0$ is the solution of the homogenized problem.

$$-\Delta u_0 = f + \Delta \bar{u} \quad\mbox{ in }\quad ]0,1[^d$$ (5.2)

As trial functions for $u_0$ we use specially tailored wavelets with homogeneous and/or periodic BC. The next section contains more details on such wavelets and how to use them.

Now, the difficult thing is to find a numerical approximation to $\bar{u}$. This function should be as smooth as possible, as we apply the discrete operator to it for the calculation of the modified right hand side in (5.2). In sections 5.2 and 5.3 it is briefly explained how $\bar{u}$ is determined.