| Abstract: |
| Many disciplines of practical mathematics, such as elasticity, fluid dynamics, reaction-diffusion processes, geophysics, chemical reactor theory, and so on, have singular perturbation problems (SPPs). It is interesting to note that the SPPs solution demonstrates a multi-scale character, that is, there are thin layers where the solution varies quickly, while away from the layer(s) the solution varies slowly. So, due to the thin layer(s) in the solution of SPPs, there are many computational difficulties in the numerical solution of SPPs.
To address this class of problems, numerous variations of discontinuous Galerkin (DG) methods have been used. The DG technique is a popular finite element technique that uses a completely discontinuous polynomial space for numerical solution and test functions. A key ingredient of the DG methods is the suitable design of numerical fluxes to obtain high-order accuracy and stable schemes. Many stunning qualities have contributed to the rise in popularity of these technologies in recent years. Allowing for arbitrary domain decompositions, high-order numerical precision, total freedom in modifying the polynomial degrees in each element independent of the neighbours, and an extremely local data structure resulting in excellent parallel efficiency are some of the features of the DG methods.
In this talk, We propose a direct discontinuous Galerkin (DDG) methods for two-paramter singular perutrbation problems. We have shown that method is uniformly convergent with the order $k$ in energy norm, where $k$ is the degree of piecewise polynomial in finite element space. We have presented numerical result to verify our theoretical findings. |
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