A new basis for PHT-splines

• We make analysis of the previously constructed basis functions of PHT-splines and observe a decay phenomenon of the basis functions under certain refinement of T-meshes.
• We propose a new basis consisting of a set of local tensor product B-splines for PHT-splines which overcomes the decay phenomenon.
• Some examples are provided for solving numerical PDEs with the new basis, and comparison is made between the new basis and the original basis. Experimental results suggest that the new basis provides better numerical stability in solving numerical PDEs.

PHT-splines (polynomials splines over hierarchical T-meshes) are a generalization of B-splines over hierarchical T-meshes which possess a very efficient local refinement property. This property makes PHT-splines preferable in geometric processing, adaptive finite elements and isogeometric analysis. In this paper, we first make analysis of the previously constructed basis functions of PHT-splines and observe a decay phenomenon of the basis functions under certain refinement of T-meshes, which is not expected in applications. We then propose a new basis consisting of a set of local tensor product B-splines for PHT-splines which overcomes the decay phenomenon. Some examples are provided for solving numerical PDEs with the new basis, and comparison is made between the new basis and the original basis. Experimental results suggest that the new basis provides better numerical stability in solving numerical PDEs.

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