We study variational mesh adaptation for axially symmetric solutions to twodimensional problems. The study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy (or regularity) and global equidistribution of the monitor function. The main results are

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... d in Theorems 4.1 and 4.2. For axially symmetric problems, it is natural to choose axially symmetric mesh adaptation. To this end, it is reasonable to use the monitor function in the form G = λ 1 (r)ere T r + λ 2 (r)e θ e T θ , where er and e θ are the radial and angular unit vectors. It is shown that when higher mesh concentration at the origin is desired, a choice of λ 1 and λ 2 satisfying λ 1 (0) < λ 2 (0) will make the mesh denser at r = 0 than in the surrounding area whether or not λ 1 has a maximum value at r = 0. The purpose can also be served by choosing λ 1 to have a local maximum at r = 0 when a Winslow-type monitor function with λ 1 (r) = λ 2 (r) is employed. On the other hand, it is shown that the traditional functional provides little control over mesh concentration around a ring r = r λ > 0 by choosing λ 1 and λ 2 . In contrast, numerical results show that the new functional provides better control of the mesh concentration through the monitor function. Two-dimensional numerical results are presented to support the analysis.