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Could I apply a Q0-discretization to, say, the Poisson equation $\Delta \phi = f$ (where by Q0 I mean piecewise constant, and thus non-continuous, elements)?

Solving this with FEM, at least as I know it, involves the gradients of the discretized function $\nabla\phi$, which in a piecewise constant discretization I would expect to be zero almost everywhere, and undefined/infinity on element edges/surfaces.

Now what I'm not sure about is if using Q0 elements together with gradients simply doesn't make sense. I could probably figure out some kind of "workaround" like calculating a "pseudo-gradient" from the value differences to neighboring cells. But I suspect this would be equivalent to using a different type of finite element altogether. Maybe anyone can give me some insight on this?

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Depends on the details.

If you think about the traditional $H^1$ conforming formulation with the bilinear form $(\nabla \phi, \nabla v)$ it obviously cannot work because the stiffness matrix would be exactly zero.

If you write this equation in a mixed form $\nabla \phi = \sigma$ and $\nabla \cdot \sigma = f$ then you can use $Q_0$ for $\phi$ if $\sigma$ uses something like the lowest order Raviart-Thomas finite elements.

In nonconforming discontinuous Galerkin formulation you would use the additional terms $$(\{\nabla u\cdot n\}, [v]) + (\{\nabla v \cdot n\}, [u]) + \gamma ([u], [v])$$ over the internal element boundaries. The last term is technically valid and nonzero for $Q_0$ so it might work although I've never tried it in practice.

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There are two ways for what you are trying to do:

  • If you approximate the gradient by finite differences involving the values at the cell midpoints, then your use of Q0 elements will lead to typical finite difference stencils.

  • You can use the common discontinuous Galerkin approach in which you are allowed finite elements of arbitrary polynomial degree (including zero) because the derivatives of the Laplace equation are imposed not just in the interior of the cells, but also as jump terms across the interfaces between cells.

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