| Abstract: |
| When using classical FEM to solve a PDE problem in a domain with a complicated or non-smooth boundary, one typically requires a mesh with a large number of elements in order to accurately capture the boundary geometry. In this talk we present a discontinuous Galerkin (dG) FEM for the solution of PDE problems in a domain with a fractal boundary (the Dirichlet Poisson problem in the Koch snowflake) in which the boundary geometry is captured exactly, by using a mesh comprising elements which themselves have fractal boundary. On such meshes the classical dG normal derivative interface terms cannot be defined in the usual way because inter-element interfaces may not have a well-defined normal vector. We show how this can be addressed by introducing proxies for the normal derivative terms, based on integrals over subsets of the mesh elements. We prove well-posedness of the resulting dG FEM and provide a partial error analysis. We also discuss practical implementation of the method and provide numerical results demonstrating its effectiveness. This is joint work with Sergio Gomez and Andrea Moiola. |
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