Scientific visualization tools tend to be flexible in some ways (e.g., for exploring isovalues) while restricted in other ways, such as working only on regular grids, or only on unstructured meshes (as used in the finite element method, FEM). Our work seeks to expose the common structure of visualization methods, apart from the specifics of how the fields being visualized are formed. Recognizing that previous approaches to FEM visualization depend on efficiently updating computed positions within a mesh, we took an existing visualization domain-specific language, and added a mesh position type and associated arithmetic operators. These are orthogonal to the visualization method itself, so existing programs for visualizing regular grid data work, with minimal changes, on higher-order FEM data. We reproduce the efficiency gains of an earlier guided search method of mesh position update for computing streamlines, and we demonstrate a novel ability to uniformly sample ridge surfaces of higher-order FEM solutions defined on curved meshes.