We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally $C^4$-smooth isogeometric spline space [34] to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the $L^2$-norm as well as to equivalents of the $H^s$-seminorms for $1 \leq s \leq 4$. In the studied case of spline degree $p=9$, the numerical results indicate in case of the Greville points a convergence of order $\mathcal{O}(h^{p-3})$ independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order $\mathcal{O}(h^{p-2})$ for all (semi)norms except for the equivalent of the $H^4$-seminorm, where the order $\mathcal{O}(h^{p-3})$ is anyway optimal.

翻译：本文提出了一种等几何配点法，用于求解平面双线性参数化多片域上的双调和方程。所开发的方法基于全局$C^4$光滑等几何样条空间[34]来逼近所考虑偏微分方程的解，并提出了两种不同的配点选择：一方面采用Greville点，另一方面采用所谓的超收敛点。多个算例展示了我们的配点法在求解平面多片域双调和方程方面的潜力，并从数值上研究了这两类配点法在$L^2$范数以及$1 \leq s \leq 4$的$H^s$半范数等价形式下的收敛行为。在样条次数$p=9$的研究案例中，数值结果表明，Greville点的收敛阶为$\mathcal{O}(h^{p-3})$，与所考虑的(半)范数无关；而超收敛点则在除$H^4$半范数等价形式之外的所有(半)范数下表现出改进的收敛阶$\mathcal{O}(h^{p-2})$，其中$H^4$半范数的$\mathcal{O}(h^{p-3})$阶收敛已是最优结果。