HDF5 is, to some extent, a filesystem on its own. By introducing B-Trees and by the way it manages blocks, it duplicates the functionality of a filesystem. When you are running your code, you are ...

To create a coarser mesh, you can set the characteristic length globally to a larger value, e.g., SetFactory("OpenCASCADE"); Mesh.CharacteristicLengthFactor = 2; Circle(1) = {0, 0, 0, 1, 0, 2*Pi}; ...

You write, that you are computing the eigenvalues of a symmetric matrix. Does the matrix have real entries? In this case all eigenvalues are real, and you can use a symmetric eigenvalue solver, which ...

You have noticed that any eigenvector of $A$ is also an eigenvector of your composed matrix. When you compute eigenvectors of your matrix, you can use them in a deflation-type preconditioner, as ...

Yes. This behavior is to be expected and normal. When you are computing with a small value for $\mathrm dx$ then, to compute the difference quotient, you are subtracting two numbers that are nearly ...

In case you can easily implement the roated grid, this is possibly the easist fix, so let me answer your question about the invariance of the convection diffusion equation under rotation of the ...

You say that the problem with complex arithmetic is that there are different ways to define the scalar product for complex vectors, compared to just one way in the real case. I think the real problem ...

Your question leads to two different considerations. First, you need to pick the right algorithm. Hence, the question if the matrices have any structure, should be considered. E.g., when the matrices ...

First of all, the term $\frac{\| A x \|}{\| x \|}$ is not constant. For example, consider a $2 \times 2$ matrix and let $w_1$ and $w_2$ be the columns of $A$. Then, $$A x = x_1 w_1 + x_2 w_2 .$$ ...

Assume you have a metric for some problem, e.g., the asymptotic convergence rate. If you can prove that there can be no algorithm that has a better value in the specific metric, then the algorithm is ...

Let us consider the abstract linear problem $$\mathcal{A} x = b \,,$$ where $\mathcal{A}$ is a linear operator and $x$ and $b$ some functions on a certain domain. To answer your question let me ...

Let me try to answer your question concerning the transport equation from a different perspective. Consider the following initial-value problem. We search for a function $v(x,t)$ such that \begin{...

The Runge-Kutta approximates the solution of the initial value problem $$\dot{y}(t) = F(t, y(t)), \quad y(0) = y_0.$$ The method expects $y_0$ and the function $F$ as input. If we set  y(t) = \...