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. |

# Updates

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

Categories: Updates

## Friday, Jan 12, 11:00AM-12:00Noon, Fretwell 315

Categories: Updates

## Friday, Dec 1st, 4:00PM-5:00PM, Fretwell 206

Categories: Updates

Professor: Louis H Kauffman, Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago |

Title:Graphical Invariants of Knots and Links |

Abstract:This talk is joint work with Qingying Deng. We generalize the signed Tutte polynomial relationship with the Kauffman bracket model of the Jones polynomial to a new polynomial defined on signed cyclic graphs (graphs with signed edges and cyclic orders at the vertices) and show how these graphs are codings for checkerboard colorable virtual links. We show how virtualization of classical links corresponds to simple operations on the planar signed cyclic graphs. We relate our new polynomial invariant to both the bracket polynomials for virtual knots and links and to the Bollobas-Riordan polynomial. |

## Friday, October 13, 11:00AM-12:00Noon, Conference room

Categories: Updates

Professor Yanlai Chen , Dept of Mathematics, University of Massachusetts, Dartmouth |

Title:Ultra-efficient Reduced Basis Method and Its Integration with Uncertainty Quantification |

Abstract: Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM solution is maintained through a rigorous a posteriori error estimator whose efficient development is critical and involves fast eigensolvers. After giving a brief introduction of the RBM/RCM, this talk will show our recent work on significantly delaying the curse of dimensionality for uncertainty quantification, and new fast algorithms for speeding up the offline portion of the RBM/RCM by around 6-fold. |

## Wednesday, August 30, 2:00PM-3:00PM, Conference room

Categories: Updates

Professor Seokchan Kim, Dept of Mathematics, ChangWon National University, Changwon, Korea |

Title:FEM to compute Numerical Solution of PDEs with Corner Singularities using SIF |

Abstract: We consider the Poisson equation with homogeneous Dirichlet boundary condition defined on non convex polygonal domain with one re-entrant corner. Solution of such equation has singular behavior near that re-entrant corner and can be expressed as a sum of the regular part and the singular part. The coefficient of the singular part is called ‘the Stress Intensity Factor. The talk is to introduce a new method to obtain an accurate numerical solution for the Poisson Equation with corner singularities using the Stress Intensity Factor. |

## Friday, March 31, 11:00AM-12:00Noon, Conference room

Categories: Updates

Ching-Shan Chou, Department of Mathematics, The Ohio State University |

Title:Cell signaling, cell morphogenesis and cell-cell communication |

Abstract: Cell-to-cell communication is fundamental to biological processes which require cells to coordinate their functions. In this talk, we will present the first computer simulations of the yeast mating process, which is a model system for investigating proper cell-to-cell communication. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the alpha-factor protease Bar1, and the regulation of sensing sensitivity. |

## Wednesday, March 29, 3:30-4:30PM, Fretwell 315

Categories: Updates

## Friday, March 17, 10:00-11:00AM, Conference room

Categories: Updates

## Monday, January 23rd, 3:30-4:30 PM, Conference room

Categories: Updates

Valery Romanovski, Center for Applied Mathematics and Theoretical Physics |

Title: Some algebraic tools for investigation of systems of ODEs |

Abstract: We give an introduction to algorithms of the elimination theory and methods for solving polynomial systems and show how they can be used for the qualitative investigation of autonomous systems of ordinary differential equations. We then apply them to study the May-Leonard system which models some ecological and chemical processes. |

## Thursday, January 5th, 11:00AM in Conference room

Categories: Updates

Duk-Soon Oh, Department of Mathematics, Rutgers University |

Title: Domain Decomposition Methods |

Abstract: |