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For implicit methods, you can achieve order $2s$ with $s$ stages. Note that this result is the same if one considers the simpler problem of numerical integration (quadrature). Update as of 2024: a 16 …

I know you asked about FE methods, but even simple finite difference methods can deal with this singularity reasonably well. Here is a solution of a 1D problem computed with a 3-point centered differ …

To add to Fredrik Johansson's answer: A nice history of algorithms for computing Gauss quadrature rules can be found in this SIAM News article by Alex Townsend. Therein, it is stated that the "final …

Regarding 1 and 2: Perhaps the main reason for considering only bounded intervals is that numerical analysts are interested in provably (pointwise) convergent schemes. At least for traditional method …

As noted already in the comments, your boundary conditions seem off. Note that generically for a second-order BVP one expects to impose only two boundary conditions; you have 4. Once you’re sure you …

The reason you don't get conservation is that you've used the product rule before discretizing, so conservation would require an exact cancellation of truncation errors in the different product terms …

A bound of the kind you're looking for is not possible, and it doesn't even matter what interpolation points you use (much less which particular polynomial basis you choose to work with, since that ha …

For numerical analysis, I can recommend: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems by R. J. LeVeque. Spectral methods with M …

There is a whole subfield of applied mathematics devoted to developing ODE solvers and understanding their properties. Consequently, there are thousands of relevant papers, and not much more can be s …

I believe you are asking the following: What is the minimum number of evaluations of $f$ required to approximate $f^{(k)}$ to order of accuracy $p$? In fact, this is a homework problem I often g …

Obviously, within the realm of piecewise-smooth functions one can find examples where any derivative-based approach fails. I believe you're looking for the term "derivative free optimization". He …

For a recent reference that includes efficient computational techniques developed in the last few years, see Chapters 26-27 of L. N. Trefethen's Approximation Theory and Approximation Practice. You c …

I believe the intended reference regarding parabolic PDEs is: Fritz, John. On integration of parabolic equations by difference methods: I. Linear and quasi-linear equations for the infinite inter …

If I understand correctly, what you're doing amounts to: Starting with initial data $\psi(x,0)$ that is represented as a truncated Fourier series. Computing the exact (up to roundoff errors) time ev …

You have assumed an equidistant grid in one dimension, but the answer below can be formulated (and is true) for general grids in any number of dimensions. You also haven't specified the boundary cond …