线形方程求解器：Direct 和 Iterative Solvers
2013-11-12 20:26阅读：
1. 直接求解器
Direct Methods
The direct solvers used by COMSOL are the
MUMPS,
PARDISO, and
SPOOLES solvers. All of the solvers are based
on
LU decomposition.
These solvers will all arrive at the same answer for all
well-conditioned finite element problems, which is their biggest
advantage, and can even solve some quite ill-conditioned problems.
From the point of view of the solution, it is irrelevant which one
of the direct solvers you choose, as they will return the same
solution. The direct solvers differ primarily in their relative
speed. The MUMPS, PARDISO, and SPOOLES solvers can ea
ch take advantage of all of the processor cores on a single
machine, but PARDISO tends to be the fastest and SPOOLES the
slowest. SPOOLES also tends to use the least memory of all of the
direct solvers. All of the direct solvers do require a lot of RAM,
but MUMPS and PARDISO can store the solution
out-of-core, which means that they can offload some
of the problem onto the hard disk. The MUMPS solver also supports
cluster computing, allowing you to use more memory than is
typically available on any single machine.
If you are solving a problem that does not have a solution (such as
a structural problem with loads, but without constraints) then the
direct solvers will still attempt to solve the problem, but will
return an error message that looks similar to:
Failed to find a solution. The relative residual (0.06) is greater
than the relative tolerance. Returned solution is not
converged.
If you get this type of error message, then you should check to
make sure that your problem is correctly constrained.
2. 迭代求解器
Iterative Methods
The iterative solvers in COMSOL encompass a variety of approaches,
but they are all conceptually quite simple to understand at their
highest level, being essentially similar to a
conjugate gradient method. Other variations include
the
generalized minimum residual method and
the
biconjugate gradient stabilized method, and
there are many variations on these, but they all behave
similarly.
Contrary to direct solvers, iterative methods approach the solution
gradually, rather than in one large computational step. Therefore,
when solving a problem with an iterative method, you can observe
the error estimate in the solution decrease with the number of
iterations. For well-conditioned problems, this convergence should
be quite monotonic. If you are working on problems that are not as
well-conditioned, then the convergence will be slower. Oscillatory
behavior of an iterative solver is often an indication that the
problem is not properly set up, such as when the problem is not
sufficiently constrained. A typical convergence graph for an
iterative solver is shown below:
By default, the model is considered converged when the estimated
error in the iterative solver is below 10
-3. This is
controlled in the Solver Settings window:
This tolerance can be made looser, for faster solutions, or
tighter, for greater accuracy on the current mesh. The tolerance
must always be greater than a number that depends on the machine
precision (2.22×10
-16) and the condition number (which
is problem dependent). However there is usually no point in making
the tolerance too tight since the inputs to your model, such as
material properties, are often not accurate to more than a couple
of digits. If you are going to change the relative tolerance, we
generally recommend making the tolerance tighter in increments of
one order of magnitude and comparing solutions. Keep in mind that
you are only solving to a tighter tolerance on the mesh that you
are currently using, and it is often more reasonable to refine the
mesh.
The big advantage of the iterative solvers is their memory usage,
which is significantly less than a direct solver for the same sized
problems. The big disadvantage of the iterative solvers is that
they do not always “just work”. Different physics do require
different iterative solver settings, depending on the nature of the
governing equation being solved.
Luckily, COMSOL already has built-in default solver settings for
all predefined physics interfaces. COMSOL will automatically detect
the physics being solved as well as the problem size, and choose
the solver — direct or iterative — for that case. The default
iterative solvers are chosen for the highest degree of robustness
and lowest memory usage, and do not require any interactions from
the user to set them up.
Concluding Thoughts on Direct and Iterative Solution
Methods
When solving the systems of linear equations of a simulation,
COMSOL will automatically detect the best solver without requiring
any user interaction. The direct solvers will use more memory than
the iterative solvers, but can be more robust. Iterative solvers
approach the solution gradually, and it is possible to change the
convergence tolerance, if desired.
