Numer. Math. (2011) 118:531–548 DOI 10.1007/s00211-010-0353-0 Numerische Mathematik Spline spaces on TR-meshes with hanging vertices Larry L. Schumaker · Lujun Wang Received: 21 May 2010 / Revised: 19 October 2010 / Published online: 14 December 2010 © Springer-Verlag 2010 Abstract Polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles are studied for the C 0 case. These include triangulations with hanging vertices as well as T-meshes. In addition to dimension formulae, explicit basis func- tions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated. Mathematics Subject Classification (2000) 41A15 · 65D07 · 65N30 1 Introduction Spaces of bivariate piecewise polynomials (splines) defined on meshes consisting of triangles or rectangles play an important role in approximation theory and numerical analysis, particularly in the finite-element method. For many applications it is impor- tant to create local refinements of the mesh. Standard local refinement algorithms are designed to avoid introducing hanging vertices. However, allowing such vertices leads to much simpler refinement algorithms, and produces fewer subtriangles or subrec- tangles. Consequently, meshes with hanging vertices have started to attract attention in the finite element literature, see e.g. [1–3, 6, 14]. Spline spaces defined on meshes with hanging vertices do not seem to have been treated in the current spline literature, with the exception of some dimension results for very special meshes of rectangles called T-meshes, see [4, 5, 11] and references therein. Our aim in this paper is to initiate the study of such spaces by discussing dimension, the construction of basis functions, and approximation power. In this paper we focus L. L. Schumaker (B ) · L. Wang Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA e-mail: larry.schumaker@vanderbilt.edu L. Wang e-mail: lujun.wang@vanderbilt.edu 123