Domain decomposition methods: Difference between revisions

Revision as of 06:19, 18 February 2025

Latest revision as of 14:45, 13 June 2025

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==Example 1: 1D Linear BVP==

<math display="block">\begin{cases}

<math> u''(x)-u(x)=0 </math><br>

<math> u''(0x) =0, u(1x)=1, </math><br>\\

u(0) = 0, \\

The exact solution is:<br>

u(1) = 1.

<math> u(x)=\frac{e^x-e^{-x}}{e^{1}-e^{-1}} </math><br>

\end{cases} </math>

Subdivide the ___domain into two subdomains, one from <math>\left[0,\frac{1}{2}\right]</math> and another from <math>\left[\frac{1}{2},1\right]</math>. In the left subdomain define the interpolating function <math> v_1(x) </math> and in the right define <math> v_2 (x) </math>. At the interface between these two subdomains the following interface conditions shall be imposed:<br>

The exact solution is:<br>

<math> v_1\left(\frac{1}{2}\right)=v_2 \left(\frac{1}{2}\right) </math><br>

<math display="block"> v_1'\leftu(x)=\frac{1e^x-e^{-x}{2}\right)=v_2'\left(\frac{e^{1}-e^{2-1}\right)} </math><br>

Subdivide the ___domain into two subdomains, one from <math>\left[0,\fractfrac{1}{2}\right]</math> and another from <math>\left[\fractfrac{1}{2},1\right]</math>. In the left subdomain define the interpolating function <math> v_1(x) </math> and in the right define <math> v_2 (x) </math>. At the interface between these two subdomains the following interface conditions shall be imposed:<br>

<math display="block">\begin{align}

<math> v_1{\left(\frac{1}{2}\right)} &=v_2 v_2{\left(\frac{1}{2}\right)} </math><br>\\

v_1'{\left(\frac{1}{2}\right)} &= v_2'{\left(\frac{1}{2}\right)}

\end{align}</math>

Let the interpolating functions be defined as:<br>

<math display="block">\begin{align}

<math> v_1 (x) =\sum_{n=0}^{N} u_{n} T_n (y_1(x)) </math><br>

<math> v_2 v_1(x) &= \sum_{n=0}^{N} u_{n+N} T_n (y_2y_1(x)) </math><br>\\

<math> v_1 v_2(x) &= \sum_{n=0}^{N} u_{n+N} T_n (y_1y_2(x)) </math><br>\\

<math> y_1(x)=4x-1 </math><br>

<math> y_2 y_1(x) &= 4x-31 </math><br>\\

y_2(x) &= 4x-3

Where <math> T_n (y) </math> is the nth cardinal function of the chebyshev polynomials of the first kind with input argument y.<br>

\end{align} </math>

If N=4 then the following approximation is obtained by this scheme:<br>

Where <math> T_n (y) </math> is the nth cardinal function of the chebyshevChebyshev polynomials of the first kind with input argument y.<br>

<math> u_1 =0.06236 </math><br>

<math> u_2 =0.21495 </math><br>

If ''N''=4 then the following approximation is obtained by this scheme:<br>

<math> u_3 =0.37428 </math><br>

<math display="block">\begin{align}

<math> u_4 =0.44341 </math><br>

u_1 &= 0.06236, &

<math> u_5 =0.51492 </math><br>

u_2 &= 0.21495, \\

<math> u_6 =0.69972 </math><br>

u_3 &= 0.37428, &

<math> u_7 =0.90645 </math><br>

u_4 &= 0.44341, \\

This was obtained with the following MATLAB code. <br>

u_5 &= 0.51492, &

u_6 &= 0.69972, \\

u_7 &= 0.90645.

\end{align}</math>

This was obtained with the following MATLAB code. <br>

<syntaxhighlight lang="matlab">

clear all

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uex = (exp(x)-exp(-x))./(exp(1)-exp(-1));

</syntaxhighlight>

==See also==

* [[Multigrid method]]

== Related Books ==

{{refbegin}}

* Barry Smith, Petter Bjørstad, and William Gropp: ''Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations'', Cambridge Univ. Press, ISBN 0-521-49589-X (1996).

{{refend}}

==See also==

*[[Multigrid method]]

{{Numerical PDE}}

== External links ==

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| archive-date = 2021-01-26

}}

{{Numerical PDE}}

{{DEFAULTSORT:Domain Decomposition Methods}}