We present a novel isogeometric collocation method for solving the Poisson's and the biharmonic equation over planar bilinearly parameterized multi-patch geometries. The proposed approach relies on the use of a modified construction of the C^s-smooth mixed degree isogeometric spline space [20] for s=2 and s=4 in case of the Poisson's and the biharmonic equation, respectively. The adapted spline space possesses the minimal possible degree p=s+1 everywhere on the multi-patch domain except in a small neighborhood of the inner edges and of the vertices of patch valency greater than one where a degree p=2s+1 is required. This allows to solve the PDEs with a much lower number of degrees of freedom compared to employing the C^s-smooth spline space [29] with the same high degree p=2s+1 everywhere. To perform isogeometric collocation with the smooth mixed degree spline functions, we introduce and study two different sets of collocation points, namely first a generalization of the standard Greville points to the set of mixed degree Greville points and second the so-called mixed degree superconvergent points. The collocation method is further extended to the class of bilinear-like G^s multi-patch parameterizations [26], which enables the modeling of multi-patch domains with curved boundaries, and is finally tested on the basis of several numerical examples.

翻译：我们提出了一种新颖的等几何配点方法，用于求解在平面双线性参数化多片几何域上的泊松方程和双调和方程。所提方法依赖于使用改进构造的C^s光滑混合次数等几何样条空间[20]，其中对于泊松方程和双调和方程分别取s=2和s=4。该改进样条空间在多片域上除内边和价数大于一的顶点附近小邻域外，均采用最小可能次数p=s+1；而在这些邻域内则需采用p=2s+1的高次数。与在整个域上均采用相同高次数p=2s+1的C^s光滑样条空间[29]相比，该方法能以更少的自由度求解偏微分方程。为使用光滑混合次数样条函数进行等几何配点，我们引入并研究了两类不同的配点集：其一是标准Greville点向混合次数Greville点集的推广，其二是所谓的混合次数超收敛点。该配点方法进一步扩展至双线性类G^s多片参数化[26]情形，从而能够对具有曲线边界的多片域进行建模，最终通过若干数值算例验证了方法的有效性。