Singular PerturbationΒΆ

Git reference: Examples singular perturbation.

We solve a singularly perturbed elliptic problem that exibits a thin anis-tropic boundary layer that is difficult to solve. This examples demonstrates how the anisotropic refinements can save a big amount of degrees of freedom.

The computational domain is the unit cube (0, 1)^3, and the equation solved has the form :

(1)\begin{eqnarray*} - \Delta u + K^2 u &= K^2 &\hbox{ in }\Omega \\ u &= 0 &\hbox{ on }\partial\Omega \end{eqnarray*}

The boundary conditions are homegeneous Dirichlet. The right-hand side is chosen to keep the solution u(x,y,z) \approx 1 inside the domain. Here, we choose K^2 = 10^4 but everything works for larger values of K as well.

It is quite important to perform the initial refinements towards the boundary, thus providing a better initial mesh for adaptivity and making convergence faster.

Convergence graphs:

../../../_images/singpert-aniso-conv.png ../../../_images/singpert-aniso-conv-time.png

Solution and hp-mesh:

../../../_images/singpert-aniso-sln.png ../../../_images/singpert-aniso-order.png

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