Automatic Adaptivity for Evolutionary Problems Based on the Rothe's Method

P. Solin, K. Segeth, I. Dolezel, J. Cerveny, L. Dubcova, P. Kus

We present a new method for automatic adaptivity for time-dependent PDEs that
includes simultaneous mesh refinement and coarsening. The technique is based 
on a combination of our recently developed multi-mesh FEM [2,3] with the 
classical Rothe's method. The Rothe's method performs semi-discretization 
in time only while leaving the spatial variables continuous (as opposed to 
the Method of Lines).

In this way, a time-dependent PDE translates into several time-independent
ones in every time step. The actual number of the time-independent PDEs
depends on the accuracy of the time integration scheme -- for example,
one has to solve one time-independent PDE per time step when using the
backward Euler method. Each of these time-independent PDEs is solved
adaptively until a prescribed accuracy is reached, starting from
a suitable coarse mesh (master mesh). The multi-mesh FEM is used for 
simultaneous treatment of approximations on different meshes. The 
technique of arbitrary-level hanging nodes [1] is employed to prevent
conflicting refinements in the system of meshes.

If no a-priori knowledge of the solution is available,
we use a uniform master mesh. Otherwise, the master mesh is
chosen to be coarser in parts of the computational domain where the
solution does not exhibit any problematic phenomena.
In particular, note that the final mesh (or system of meshes in a coupled
problem) changes at the end of every time step -- in other words, an effect
that can be called "simultaneous mesh refinement and coarsening"
takes place naturally. In this way, spatial discretization error 
can be practically eliminated in every time step. Numerical 
examples are presented.

References

[1] P. Solin, J. Cerveny, I. Dolezel: Arbitrary-Level Hanging Nodes and Automatic
    Adaptivity in the hp-FEM, MATCOM, in press, doi:10.1016/j. matcom.2007.02.011.

[2] P. Solin, J. Cerveny, L. Dubcova: Adaptive Multi-Mesh $hp$-FEM for Linear
    Thermoelasticity, submitted to CMAME, November 2007.

[3] P. Solin, J. Cerveny, L. Dubcova, I. Dolezel: Multi-Mesh hp-FEM for Thermally 
    Conductive Incompressible Flow. In: Proceedings of ECCOMAS Conference COUPLED 
    PROBLEMS 2007 (M. Papadrakakis, E. Onate, B. Schrefler Eds.), CIMNE, Barcelona.