Abstract
In this work, a coupling of adaptive element free Galerkin method with variational multiscale element free Galerkin method has been proposed in solving two-dimensional elliptical ring solitons. Meshless method is suitable in incorporating into adaptive refinement as the additional nodes can be freely inserted at the steep gradient regions without the reconstruction of geometry as in finite element method. Hence, the advantageous of adaptive refinement technique and variational multiscale method are inherited in this proposed method in order to obtain higher accuracy solution.The performance of the proposed method is first validated with solving Burgers' equation. The obtained numerical solutions are more accurate than the standard element free Galerkin method. Besides, this coupling method is then extended to solve two-dimensional elliptical ring solitons. The numerical results are found to be in good agreement with the numerical solutions that exist in literature. Therefore, this confirms the ability of the proposed method to deal with the large gradient problems and sine-Gordon equation.
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