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In this paper we introduce general transfer operators between high-order andlow-order refined finite element spaces that can be used to couple high-orderand low-order simulations. Under natural restrictions on the low-order refinedspace we prove that both the high-to-low-order and low-to-high-order linearmappings are conservative, constant preserving and high-order accurate. Whilethe proofs apply to affine geometries, numerical experiments indicate that theresults hold for more general curved and mixed meshes. These operators alsohave applications in the context of coarsening solution fields defined onmeshes with nonconforming refinement. The transfer operators for $H^1$ finiteelement spaces require a globally coupled solve, for which robust and efficientpreconditioners are developed. We present several numerical results confirmingour analysis and demonstrate the utility of the new mappings in the context ofadaptive mesh refinement and conservative multi-discretization coupling.

Conservative and accurate solution transfer between high-order and low-order refined finite element spaces