We extend a parameter-free method for achieving high-order accuracy on polygonally approximated curved boundaries [3] in several ways. We extend [3] using the idea of the shifted boundary method [1, 2]. We apply this here both to improve accuracy of methods for curved boundaries, but also primarily to solve coupled system interactions with time-dependent boundaries between the sub-systems. Dirichlet problems on a smooth domain are frequently discretized using a polygonal approximation of the domain, causing an error that is suboptimal for piecewise-quadratic and higher-order approximations [3]. For problems with divergence constraints, the suboptimality is much worse; it has been observed [4, 5] that convergence in Lipschitz norm fails when using finite elements that are exactly divergence free. We show that a simple mesh modification avoids this flaw. Many industrial problems can be posed using the shifted boundary method. For example, flow past a vibrating cylinder can be approximated using a polygonal approximation of the cylinder. The various positions of the cylinder can be chosen outside of the mesh used for the flow problem, so the mesh can remain fixed throughout the simulation. Dirichlet boundary conditions on the cylinder can be posed with integral terms representing the boundary conditions accurately, without any penalty terms [3]. To solve flow problems, it will be necessary to extend [3] to such simulations. The interaction of vibrating cylinders and vortex dynamics is a subject of significant interest [6]. REFERENCES [1] N. M. Atallah, C. Canuto, and G. Scovazzi. Computer Methods in Applied Mechanics and Engineering, 358:112609, 2020. [2] N. M. Atallah, C. Canuto, and G. Scovazzi. Computer Methods in Applied Mechanics and Engineering, 372:113341, 2020. [3] T. Dupont, J. Guzm ́an, and L. R. Scott. Journal of Numerical Mathematics, published online, 2024. [4] I. G. Gjerde and L. R. Scott. ACSE, 2(3):295–319, 2024. [5] L. R. Scott and T. Tscherpel. Derivative constraints on finite element functions due to boundary approximation. in preparation, 2025. [6] C. H. K. Williamson and R. Govardhan. Annu. Rev. Fluid Mech., 36:413–455, 2004.