A modified bubble placement method and its application in solving elliptic problem with discontinuous coefficients adaptively

ABSTRACT

To solve the elliptic problem with discontinuous coefficients adaptively and efficiently which has strong singularity, one of the key steps is to generate mesh adaptively near region with discontinuous coefficients according to an a posteriori error estimator. For bubble placement method, the previous interaction force function used to obtain a high-quality nodes set is only determined by the ratio ω of the real and desired distance between bubbles, which is not suitable to generate mesh with a very small size. In this study, the force function is modified and it is determined both by the ratio ω and the sizes of bubbles. Meanwhile, in consideration of the quality of nodes and the computational efficiency, the equation of motion used to calculate the positions of bubbles is also modified. Moreover, a constrained bubble-type local mesh generation (BLMG) method is developed. The modified node placement method and constrained BLMG strategy are applied to solve elliptic problem with discontinuous coefficients adaptively. Several numerical experiments are reported to demonstrate that the adaptive method can be applied to various kinds of complicated geometries, and the triangles remain very well shaped at any particular refinement levels, even though the mesh size varies by several orders of magnitude.

KEYWORDS:

• Adaptive finite element method
• elliptic problem
• discontinuous coefficient
• force function
• constrained local mesh generation

2010 AMS SUBJECT CLASSIFICATIONS:

• 35R05
• 65N50
• 65M50
• 74H15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by National Natural Science Foundation of China (No: 11471262, 11501450), and the Graduate Starting Seed Fund of Northwestern Polytechnical University (No: Z2015032).