Weixiong, Have a look at step-38 (https://www.dealii.org/8.5.0/doxygen/deal.II/step_38.html) for an example. We oftn assume that the domain we want solve on has (locally) the same dimension (dim) as the space it is embedded in (spacedim). However, this must not be true and we can solve PDEs also on lower dimensional surfaces as step-38 shows.
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