  We present a distributed parallel mesh curving method for virtual geometry.
The main application is to generate large-scale curved meshes on complex
geometry suitable for analysis with unstructured high-order methods.
Accordingly, we devise the technique to generate geometrically accurate meshes
composed of high-quality elements. To this end, we advocate for degree
continuation on a penalty-based second-order optimizer that uses global tight
tolerances to converge the distortion residuals. To reduce the method memory
footprint, waiting time, and energy consumption, we combine three main
ingredients. First, we propose a matrix-free GMRES solver pre-conditioned with
successive over-relaxation by blocks to reduce the memory footprint three
times. We also propose an adaptive penalty technique, to reduce the number of
non-linear iterations. Third, we propose an indicator of the required linear
solver tolerance to reduce the number of linear iterations. On thousands of
cores, the method curves meshes composed of millions of quartic elements
featuring highly stretched elements while matching a virtual topology.
