Isogeometric $C^1$ mortar method

TL;DR

The paper tackles the biharmonic equation on multi-patch domains by enforcing $C^1$ continuity across patch interfaces in a weak, mortar-based framework. It selects a $p$-degree primal spline space and pairs it with a $p-2$ multiplier space, incorporating a vertex-smoothing strategy to guarantee discrete inf-sup stability. The authors prove optimal a priori error estimates under a discrete inf-sup condition and validate the theory with numerical experiments on complex patch domains, demonstrating optimal rates in patchwise $H^2$, $H^1$, $L^2$, and $L^{\infty}$ norms. This approach enables robust, high-order IgA discretizations for plate-like problems on complex multi-patch geometries, with practical impact for CAD-integrated PDE solvers and high-regularity simulations.

Abstract

We present an isogeometric mortar method for the discretization of the biharmonic equation posed on multi-patch domains. We assume only $C^0$-conformity at interfaces and employs a mortar approach to weakly enforce $C^1$-continuity across patch interfaces. Discrete inf-sup stability is ensured by selecting a Lagrange multiplier space consisting of splines of degree reduced by two compared to the primal space, with increased smoothness or merged elements near vertices. We prove optimal a priori error estimates and confirm the theoretical findings with a series of numerical experiments.

Figures (12)

• Figure 1: The dual basis $\widetilde{\lambda}_{i,j}^p$ appearing in \ref{['eq:bivcappi']} has a different support depending of $i,j$. For the indices depicted in blue, that is $\widetilde{\lambda}_{1,1}^p$, $\widetilde{\lambda}_{n_1,1}^p$, $\widetilde{\lambda}_{1,n_2}^p$ and $\widetilde{\lambda}_{n_1,n_2}^p$, the support is the corresponding vertex. For the green indices the supports are proper intervals of the boundary edges $\partial \widehat{\Omega}$. For the indices in black and red, the support of $\widetilde{\lambda}_{i,j}^p$ is that of the corresponding basis function $B_{i,j}^p$.
• Figure 2: Degrees of freedom for the trace space $W_\ell$ (green).
• Figure 3: Top: basis functions of the space $\mathcal{S}^3_{0,\partial}$ on the knot vector $\Xi = \{0, 0, h, \dots, 7h, 1, 1\}$. Bottom: basis functions of the space $\mathcal{S}^1_{\textsc{m}}$ on the knot vector $\Xi_{\textsc{m}} = \{0, 0, 2h, \dots, 6h, 1, 1\}$
• Figure 4: Physical domains.
• Figure 5: Relative error for the square domain with $C^2$-constraints at the vertices.

Theorems & Definitions (46)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof