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README.md

Space and time convergence tests

Author: Riccardo Tosi

Kratos version: 9.0

Source files: section 2.7 of [1]

We solve the transient convection diffusion equation $\frac{\partial \phi}{\partial t} + v \cdot \nabla \phi + \phi \nabla \cdot v - \nabla \cdot k \nabla \phi = f$ and validate its reference implementation. We refer to section 2.7.1 of [1] for details.

We validate the implementation by computing the $l^2$ norm of the error as $\varepsilon = \sqrt{\int_{\Omega} (\phi - \phi_h)^2 }$ , where $h$ is the mesh size, $\phi$ and $\phi_h$ are the analytic and FEM solutions, respectively.

The figure shows that the error $\varepsilon$ converges as expected for both quasi-static ASGS and quasi-static OSS.

We solve the transient convection diffusion equation $\frac{\partial \phi}{\partial t} + v \cdot \nabla \phi + \phi \nabla \cdot v - \nabla \cdot k \nabla \phi = f$ and validate its reference implementation. We refer to section 2.7.2 of [1] for details.

The analytic solution at time $t$ is $\phi(t) = x - \sin(t)$ . Therefore, it is possible to compute the $l^2$ norm of the error as $\varepsilon = \sqrt{\int_{\Omega} (\phi - \phi_h)^2 }$ , where $h$ is the mesh size, $\phi$ and $\phi_h$ are the analytic and FEM solutions, respectively. It is expected to obtain an order four accuracy for the Runge-Kutta 4 time integration scheme.

The figure shows that the time accuracy is of order four, as expected.