Compatible remeshing provides meshes with common connectivity structures. The existing compatible remeshing methods usually suffer from high computational cost or poor quality. In this paper, we present a fast method for computing compatible meshes with high quality. Given two closed, oriented, and topologically equivalent surfaces and a sparse set of corresponding landmarks, we first compute a bijective inter-surface mapping, from which compatible meshes are generated. We then improve the remeshing quality by using a volume-enhanced optimization. In contrast to previous work, our method designs a fast volume-enhanced improvement procedure that directly reduces the isometric distortion of the map between the compatible meshes. Our method also tries to preserve the shapes of the input meshes by projecting the vertices of the compatible meshes onto the input surfaces. Central to this approach is the use of the monotone preconditioned conjugate gradient method, which minimizes the energies effectively and efficiently. Compared with state-of-the-art methods, our method performs about one order of magnitude faster with better remeshing quality. We demonstrate the efficiency and efficacy of our method using various model pairs.