From: Konstantin Ladutenko Date: Wed, 10 May 2017 18:09:23 +0000 (+0300) Subject: add note on discontinuous coefficient X-Git-Tag: v9.0.0-rc1~1607^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f473599b15d62fa15846e2d8158420eec4be106e;p=dealii.git add note on discontinuous coefficient --- diff --git a/examples/step-6/doc/results.dox b/examples/step-6/doc/results.dox index b41f3cf9f2..890eaa33e6 100644 --- a/examples/step-6/doc/results.dox +++ b/examples/step-6/doc/results.dox @@ -31,7 +31,7 @@ Cycle 7: -As intended, the number of cells roughly doubles in each cycle. The +As intended, the number of cells roughly doubles in each cycle. The number of degrees is slightly more than four times the number of cells; one would expect a factor of exactly four in two spatial dimensions on an infinite grid (since the spacing between the degrees @@ -484,7 +484,43 @@ step-49, step-53, and step-54, among other tutorial programs that cover the issue. Information on curved domains can also be found in the documentation module on @ref manifold "Manifold descriptions". - +Why does it make sense to choose a mesh that tracks the internal +interface? There are a number of reasons, but the most essential one +comes down to what we actually integrate in our bilinear +form. Conceptually, we want to integrate the term $A_{ij}^K=\int_K +a(\mathbf x) \nabla \varphi_i(\mathbf x) \nabla \varphi_j(\mathbf x) ; dx$ as the +contribution of cell $K$ to the matrix entry $A_{ij}$. We can not +compute it exactly and have to resort to quadrature. We know that +quadrature is accurate if the integrand is smooth. That is because +quadrature in essence computes a polynomial approximation to the +integrand that coincides with the integrand in the quadrature points, +and then computes the volume under this polynomial as an approximation +to the volume under the original integrand. This polynomial +interpolant is accurate if the integrand is smooth on a cell, but it +is usually rather inaccurate if the integrand is discontinuous on a +cell. + +Consequently, it is worthwhile to align cells in such a way that the +interfaces across which the coefficient is discontinuous are aligned +with cell interfaces. This way, the coefficient is constant on each +cell, following which the integrand will be smooth, and its polynomial +approximation and the quadrature approximation of the integral will +both be accurate. Note that such an alignment is common in many +practical cases, so deal.II provides a number of functions (such as +@ref GlossMaterialId "material_id") to help manage such a scenario. +Refer to step-28 and step-46 for examples of how material id's can be +applied. + +Finally, let us consider the case of a coefficient that has a smooth +and non-uniform distribution in space. We can repeat once again all of +the above discussion on the representation of such a function with the +quadrature. So, to simulate it accurately there are a few readily +available options: you could reduce the cell size, increase the order +of the polynomial used in the quadrature formula, select a more +appropriate quadrature formula, or perform a combination of these +steps. The key is that providing the best fit of the coefficient's +spatial dependence with the quadrature polynomial will lead to a more +accurate finite element solution of the PDE.

Playing with the regularity of the solution