Professor:Jae Woo JEONG, Department of Mathematics, Miami University |

Title:Numerical Methods for Biharmonic Equations on non-convex Domains |

Abstract: Several methods constructing C1-continuous basis functions have been introduced for the numerical solutions of fourth-order partial differential equations. However, implementing these C1-continuous basis functions for biharmonic equations is complicated or may encounter some difficulties. In the framework of IGA (IsoGeometric Analysis), it is relatively easy to construct highly regular spline basis functions to deal with high order PDEs through a single patch approach. Whenever physical domains are non convex polygons, it is desirable to use IGA for PDEs on non-convex domains with multi-patches. In this case, it is not easy to make patchwise smooth B-spline functions global smooth functions.In this talk, we propose two new approaches constructing C1-continuous basis functions for biharmonic equation on non-convex domain: (i) Firstly, by modifying Bezier polynomials or B-spline functions, we construct hierarchical global C1-continuous basis functions whose imple- mentation is as simple as that of conventional FEM (Finite Element Methods). (ii) Secondly, by taking advantages of proper use of the control point, weights, and NURBS (Non-Uniform Rational B-Spline), we construct one-patch C1-continuous geometric map onto an irregular physical domain and associated C1-continuous basis functions. Hence, we can avoid the difficulties aris- ing multi patch approaches. Both of the proposed methods can be easily extended to construct highly smooth basis functions for the numerical solutions of higher order partial differential equations. |

# Tuesday, January 9th, 11:00AM-12:00 noon, Conference room

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