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THE SMOOTH EXTENSION EMBEDDING METHOD
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DOI：10.1137/19M1300844.png)
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摘要
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A new approach to the solution of boundary value problems within the so-called fictitious domain method philosophy is proposed, which avoids well-known shortcomings of other methods of this type, including the need to generate extensions of the data. The salient feature of the novel method, which we refer to as SEEM (smooth extension embedding method), is that it reduces the whole boundary value problem to a linear constraint for an appropriate optimization problem formulated in a larger, simpler set containing the domain on which the boundary value problem is posed and which allows for the use of straightforward discretizations. It can also be viewed as a fully discrete meshfree method, which uses a novel class of basis functions and thus builds a bridge between fictitious domain and meshfree methods. The proposed method, in essence, computes a (discrete) extension of the solution to the boundary value problem by selecting it as a smooth element of the complete affine family of solutions of the original equations now yielding an underdetermined problem for an unknown defined in the whole fictitious domain. The actual regularity of this extension is determined by that of the analytic solution and by the choice of objective functional. Numerical experiments demonstrate that it is stable enough to efficiently solve boundary value problems on general geometries and that it produces solutions of tunable (and high) accuracy.
Keyword:
fictitious domain methods
meshfree methods
numerical solutions of boundary value problems
optimization
high order discretizations
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2.6
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5.1K
被引数:
1.8W
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