A parametrix for the surface Stokes equation

Abstract

We introduce an integral equation formulation of the surface Stokes equations, constructed using two-dimensional Stokeslets. The resulting integral equations are Fredholm integral equations of the second kind and can be discretized to high order using standard tools. Since the resulting discrete linear systems are dense, we describe and analyze a proxy shell method to construct fast direct solvers for these systems. The properties of our integral equation, and the performance of the resulting numerical scheme, are illustrated with several representative numerical examples.

Figures (10)

• Figure 1: An illustration of the local discretization. Left: the Vioreanu-Rokhlin nodes of order 4 on the standard triangle $T_0.$ The nodes are shown in blue and $T_0$ is shaded red. Right: the image of the Vioreanu-Rokhlin nodes (blue) on a patch $\Gamma_i\subset \mathbb{R}^3$ (red).
• Figure 2: An illustration of the adaptive integration strategy used in the near-field integral computation. Left: the standard triangle $T_0.$ Middle: the first level of refinement. Right: a cartoon of the refinement used for a source located at the point indicated by a star.
• Figure 3: The maximum error in the compressed approximation of the logarithmic kernel found in the test described in Section \ref{['sec:compress_test']}. It is plotted as a function of the number of proxy shells and the order of the quadrature scheme on each shell.
• Figure 4: The maximum error in the compressed approximation of the remainder kernel found in the test described in Section \ref{['sec:compress_test']}. It is plotted as a function of the number of proxy shells and the order of the quadrature scheme on each shell.
• Figure 5: The reference solution $\boldsymbol u$ used for the convergence test with errors in \ref{['fig:slant_err']}. The color indicates the norm of $\boldsymbol u$.

Theorems & Definitions (7)

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