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6

You should stick with GMRES, it is the only method that is essentially guaranteed to get a solution here. The real problem appears to be you need a better preconditioner. You could try sticking with LU but adding a diagonal mass matrix to the system with a constant multiplier that decreases with the linear residual of the system. This would allow you to get ...

4

One approach to convert this into an ODE is with index reduction methods. These allow you to convert high-index DAEs into low-index DAEs or ODEs. See section VII.2 of "Solving Ordinary Differential Equations II" by Hairer and Wanner. Consider a generic, Hessenberg index-2 DAE \begin{align} y' &= f(y, z) \\ 0 &= g(y) \end{align} ...

3

To expand on @GoHokies comment: The answer to your all your questions is basically "because $a$ is bilinear and symmetric" (necessarily so; without symmetry this approach wouldn't work). Specifically, don't think of gradients but of directional derivatives: You have a saddle point if the derivative in every direction vanishes, i.e., if the Fréchet ...

1

It's a misunderstanding that you need two different meshes: The proper way to see things is that you are using the same mesh, but different polynomial spaces for the two variables. For example, for the Stokes equation, you'd have quadratic polynomials for the velocity $\mathbf u$ and linear polynomials for the pressure $p$. Appropriate parallelization ...

1

This should be a "comment", but I don't have the credentials. I interpret the question as being about understanding the workings of the Uzawa methods, their stability and optimization, rather than how to solve the diffusion equation. The following articles are easily readable and address this: Ho et al. https://arxiv.org/abs/1510.04246 ("Accelerating the ...

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