On Riesz minimal energy problems

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• 作者：H. Harbrecht^c ; ^{helmut.harbrecht@unibas.ch} ; W.L. Wendland^a ; ^{wendland@mathematik.uni-stuttgart.de} ; N. Zorii^b ; ^{natalia.zorii@gmail.com}
• 关键词：Minimal Riesz energy problem ; External field ; Pseudodifferential operator ; Penalty method ; Simple layer boundary integral operator ; Boundary element approximation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2012
• 期刊代码：76_0022247x
• 类别：mt
• 出版时间：15 September, 2012
• 卷：393
• 期：2
• 页码：397-412
• 文件大小：737 K

摘要

In , , we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where , for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless -dimensional -manifolds , , each being charged with Borel measures with the sign prescribed. We show that the Gauss variational problem over an affine cone of Borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the Sobolev-Slobodetski space , where and . This allows the application of simple layer boundary integral operators on and, hence, a penalty approximation. A corresponding numerical method is based on the Galerkin-Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparsify the system matrix. To the discretized problem, a gradient-projection method is applied. Numerical results are presented to illustrate the approach.