156 research outputs found
On the mesh nonsingularity of the moving mesh PDE method
The moving mesh PDE (MMPDE) method for variational mesh generation and
adaptation is studied theoretically at the discrete level, in particular the
nonsingularity of the obtained meshes. Meshing functionals are discretized
geometrically and the MMPDE is formulated as a modified gradient system of the
corresponding discrete functionals for the location of mesh vertices. It is
shown that if the meshing functional satisfies a coercivity condition, then the
mesh of the semi-discrete MMPDE is nonsingular for all time if it is
nonsingular initially. Moreover, the altitudes and volumes of its elements are
bounded below by positive numbers depending only on the number of elements, the
metric tensor, and the initial mesh. Furthermore, the value of the discrete
meshing functional is convergent as time increases, which can be used as a
stopping criterion in computation. Finally, the mesh trajectory has limiting
meshes which are critical points of the discrete functional. The convergence of
the mesh trajectory can be guaranteed when a stronger condition is placed on
the meshing functional. Two meshing functionals based on alignment and
equidistribution are known to satisfy the coercivity condition. The results
also hold for fully discrete systems of the MMPDE provided that the time step
is sufficiently small and a numerical scheme preserving the property of
monotonically decreasing energy is used for the temporal discretization of the
semi-discrete MMPDE. Numerical examples are presented.Comment: Revised and improved version of the WIAS preprin
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