A H1-integrability condition of surfaces with singular parametrizations in isogeometric analysis

Singularities of a surface's geometric mapping (or parametrization) are often unavoidable, especially when a complex surface is considered. In isogeometric analysis, singularities impact the regularity of test functions. When a second-order partial differential equation, such as a Laplace-Beltrami equation, is solved, the test functions should satisfy the H1-regularity, which may be destroyed by singularities. In this paper, we consider the H1-regularity of test functions on a surface by parametrization with isolated singularities. A H1-integrability condition is presented, and we apply this condition to discuss the H1-regularity of test functions by two common singular parametrizations of surfaces such as spheres and D-patches.