In a two-dimensional space, you can introduce complex structure, which can be used to elegantly solve many problems (e.g. potential flow problems), but no analogue exists in 3 dimensions.

As mentioned in Geoff Oxberry's more complete answer, it should be noted that PETSc includes TSSUNDIALS, an interface to SUNDIALS. If you configure PETSc with the --download-sundials option (see ...

If your question is "do people sometimes use the $2$-norm to measure distances without full justification?," then the answer is probably yes. Choosing the correct norm to measure differences between ...

See @Kirill's answer, but depending on what sort of hardware you have available, you might be able to leverage parallelism to solve the problem more quickly. For example, say you had a way to evaluate ...

Update: the advice below is necessary, but was written just to expand on the comments above and isn't the actual issue with the code above. See the other answers. You have a discretized system $Ax=b$,...

Your confusion mainly seems to be with calculus, not with numerical methods. Specifically, make sure you understand the difference between an indefinite integral and a definite integral. consider ...

I assume that you are comparing multiplication with an assembled PETSc matrix with your hand-coded matrix-free method. The latter may indeed be faster, but this could be because no entries of A need ...

In some cases, (F)MG provides an algorithm with optimal properties. For instance, properly tuned FMG can solve some elliptic problems in a small number of "work units", where a work unit is defined to ...

PETSc does this. See the documentation for MatShell and Section 3.3 in the manual.

Set $e=0$. The previous $e$ there has already been "accounted for," as you have used it to correct your solution.

Yes. The reason I asked was that with a particular implementation (cray-mpich 7.0.4), in a particular environment on a particular system, the second MPI_Wait call would not return. This was likely ...

For archival purposes, note that the resolution to the problem is found in the comments below the question. Data was being read from a file containing lower-precision data.

I'd be very interested in the answer to this question as far a HPC and general scientific computing is concerned, but there have certainly been lots of nice results in computational geometry, for ...

I'm not going to answer this question in complete detail, since it might be homework, but here are some tips on how to proceed. First, decide if there's an exact solution available. If it's linear, ...

One way to think about a differential $k$-form is "something integrable over a $k$-dimensional manifold". The most familiar example would be the volume form $dx$ in $\int_0^1 x^2 dx$, but also the $x^...

As described in the documentation, one can set the 'Event' property using odeset. The following (in an m-file named ode45EventTest.m) provides a barebones example of your use case, with the trivial ...