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On the Solution of the Stokes Equation on Regions with Corners
Author(s) -
Rachh Manas,
Serkh Kirill
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21937
Subject(s) - gravitational singularity , mathematics , mathematical analysis , integral equation , stokes flow , boundary (topology) , singularity , differential equation , convergent series , geometry , flow (mathematics) , power series
The detailed behavior of solutions to Stokes equations on regions with corners has been historically difficult to characterize. The solutions to Stokes equations on regions with corners are known to develop singularities in the vicinity of corners; in particular, the solutions are known to have infinite oscillations along almost every ray that meet at the corner. While the nature of singularities for the differential equation have been analyzed in great detail, very little is known about the nature of singularities for the corresponding integral equations. In this paper, we observe that, when the Stokes equation is formulated as a boundary integral equation, the solutions are representable by rapidly convergent series of the form ∑ jc j t μ j sinβ j log t+ d j t μ j cosβ j log t, where t is the distance from the corner and the parameters μ j , β j are real, and are determined via an explicit formula depending on the angle at the corner. In addition to being analytically perspicuous, these representations lend themselves to the construction of highly accurate and efficient numerical discretizations, significantly reducing the number of degrees of freedom required for the solution of the corresponding integral equations. The results are illustrated by several numerical examples. © 2020 Wiley Periodicals LLC