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I have a partial differential equation of the form $$ \frac{d}{dt}f(x,t) = \Theta(x) f(x,t) \qquad \Theta(x) \sim \left[\frac{d^2}{dx^2} + k^2(x)\right] $$ subject to $f(x,t=0) = f_0(x)$, and $f(x=0,t) = f(x=1,t) = 0$. The function $k(x)$ is piecewise constant over $x\in[0,1]$. Call the $x$ intervals where $k$ is constant $I_j$. I would like to locally represent the solution $f$ as a sum of (say) Chebyshev polynomials within each interval $I_j$ and enforce continuity conditions at the boundaries of the intervals (continuity of $f$ and $\frac{\partial f}{\partial x}$).

Now, since the RHS operator is time independent, I could just solve this using an exponential: $f=e^{\Theta t}f_0$. My question is how I would compute the matrix elements of $\Theta$ to take into account the subdomain continuity conditions.

Alternatively, I could use a time marching method and make this a true domain decomposition method, but I still don't know how I would enforce the continuity conditions. I don't want to use a penalty method to enforce the constraints only approximately; I want them enforced as exactly as possible, and I really don't care about parallelism (so perhaps calling this DD is misleading). The literature on this probably exists, but I simply have no idea what I'm looking for.

I only give a rough description of $\Theta$, but it comes from the Helmholtz equation so it's hyperbolic, and a $2\times2$ block operator, with $f$ actually being a 2-vector of a function with its dual. This isn't terribly relevant to the question, since I just want to know how to enforce internal consistency.

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See David Kopriva - Multidomain Spectral Solution of Compressible Viscous Flows, JCP, 115, 1994. for multidomain collocation algorithm. –  Johntra Volta Dec 14 '12 at 17:00
    
What is the problem you have to enforce continuity between subdomains? Just take the Dirichlet and Neumann traces on the interface from two incident subdomain Chebyshev approximations, and match them. You get a differential algebraic equation after space discretization (DAE). Your question asks how to integrate this DAE. There are many methods you can find in the literature, e.g. Solving ODEs by Hairer-Norsett-Wanner, and maybe some new papers on the exponential integrators for DAE. –  Hui Zhang Dec 14 '12 at 23:38
    
@Hui: If I use $N$ polynomials in each subdomain, I end up with $N$ equations for their time evolution. On top of this I need two constraint equations per interface. So it seems like I would have square (uncoupled) systems $\dot{x} = Ax$ for the subdomains, and constraint equations $Bx=0$ for all the interfaces. –  Victor Liu Dec 15 '12 at 0:24

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