A Novel Paradigm for Parallel Mesh Generation
The problem we are solving is a standard convection-diffusion equation.
The region is specified by a simple description of the boundary, called
a skeleton . This skeleton
is a facsimile of the UCSD logo.
The convection is of strength
10^3 directed downward, resulting in some interesting boundary layers.
We will solve this problem using a parallel adaptive mesh paradigm which
allows the majority of the computation to be done in parallel with little
communication.
An initial mesh was created from the skeleton. This mesh was adaptively
refined to form a mesh with 5000 vertices.
This mesh is partitioned for 16 processors
according to equal error, where the error is estimated via
an a posterior error estimator. This is a small calculation and was
done on only one processor. On the right the load balance is shown without
element edges lined with black, and elements are colored according to
size.
Here is the coarse grid solution.
Each process is given the entire coarse problem;
each executes an adaptive
refinement feedback loop, but with the adaptive refinement
largely restricted to its own subregion. This step is done on all processors
with no communication.
In this example, each processor adaptively
solved a problem with approximately 20K vertices, with a mesh refined
in its region and coarse elsewhere.
The refined meshes for each processor are then combined to
form a global mesh. This mesh is generally nonconforming
along the interfaces.
The global mesh is made conforming by adjusting the vertices along
the interface; the final global
mesh has about 224k vertices. A nonconforming solution
generated from the fine parts of the solution on each processor
is used as initial guess for the
final solution on the global mesh.
On the left is the global mesh colored by size.
On the right is the global mesh colored by element error.
The final global conforming solution is generated by a special domain
decomposition algorithm, based on solving global problems on the meshes
generated on each processor. This DD solver also has very low communication
costs.
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