Isogeometric collocation on planar multi-patch domains
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We present an isogeometric framework based on collocation for solving the Poisson's equation over planar bilinearly parameterized multi-patch domains. As a discretization space for the partial differential equation a globally C^2-smooth isogeometric spline space is used, whose construction is simple and works uniformly for all possible multi-patch configurations. The basis of the C^2-smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degrees. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1,14,31]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degrees.
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