• East Asian mathematical journal
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• 제14권2호
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• pp.399-410
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• 1998

In [3], Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated-linear systems. In this work, we present computational experiments of W-cycle multigrid method. Computational experiments show that the convergence is uniform as the parameter, $\nu$, goes to 1/2.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제1권1호
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• pp.105-125
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• 1997

In this paper we consider multigrid algorithms for solving elasticity problems by using Wilson's nonconforming finite element method. We consider two types of intergrid transfer operators which is needed to define the multigrid algorithm and prove convergence of $\mathcal{W}$-cycle mutigrid algorithm and uniform condition number estimates for the variable $\mathcal{V}$-cycle multigrid preconditioner.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권2호
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• pp.65-92
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• 2019

We survey a recently developed immersed finite element method (IFEM) for the interface problems. The IFEM uses structured grids such as uniform grids, even if the interface is a smooth curve. Instead of fitting the curved interface, the bases are modified so that they satisfy the jump conditions along the interface. The early versions of IFEM [1, 2] were suboptimal in convergence order [3]. Later, the consistency terms were added to the bilinear forms [4, 5], thus the scheme became optimal and the error estimates were proven. For elasticity problems with interfaces, we modify the Crouzeix-Raviart based element to satisfy the traction conditions along the interface [6], but the consistency terms are not needed. To satisfy the Korn's inequality, we add the stabilizing terms to the bilinear form. The optimal error estimate was shown for a triangular grid. Lastly, we describe the multigrid algorithms for the discretized system arising from IFEM. The prolongation operators are designed so that the prolongated function satisfy the flux continuity condition along the interface. The W-cycle convergence was proved, and the number of V-cycle is independent of the mesh size.