  We propose and analyze a structure-preserving parametric finite element
method (SP-PFEM) to simulate the motion of closed curves governed by
area-conserved generalized mean curvature flow in two dimensions (2D). We first
present a variational formulation and rigorously prove that it preserves two
fundamental geometric structures of the flows, i.e., (a) the conservation of
the area enclosed by the closed curve; (b) the decrease of the perimeter of the
curve. Then the variational formulation is approximated by using piecewise
linear parametric finite elements in space to develop the semi-discrete scheme.
With the help of the discrete Cauchy's inequality and discrete power mean
inequality, the area conservation and perimeter decrease properties of the
semi-discrete scheme are shown. On this basis, by combining the backward Euler
method in time and a proper approximation of the unit normal vector, a
structure-preserving fully discrete scheme is constructed successfully, which
can preserve the two essential geometric structures simultaneously at the
discrete level. Finally, numerical experiments test the convergence rate, area
conservation, perimeter decrease and mesh quality, and depict the evolution of
curves. Numerical results indicate that the proposed SP-PFEM provides a
reliable and powerful tool for the simulation of area-conserved generalized
mean curvature flow in 2D.
