Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monotonicity-preserving finite element schemes based on differentiable
  nonlinear stabilization   In this work, we propose a nonlinear stabilization technique for scalar
conservation laws with implicit time stepping. The method relies on an
artificial diffusion method, based on a graph-Laplacian operator. It is
nonlinear, since it depends on a shock detector. The same shock detector is
used to gradually lump the mass matrix. The resulting method is LED, positivity
preserving, linearity preserving, and also satisfies a global DMP. Lipschitz
continuity has also been proved. However, the resulting scheme is highly
nonlinear, leading to very poor nonlinear convergence rates. We propose a
smooth version of the scheme, which leads to twice differentiable nonlinear
stabilization schemes. It allows one to straightforwardly use Newton's method
and obtain quadratic convergence. In the numerical experiments, steady and
transient linear transport, and transient Burgers' equation have been
considered in 2D. Using the Newton method with a smooth version of the scheme
we can reduce 10 to 20 times the number of iterations of Anderson acceleration
with the original non-smooth scheme. In any case, these properties are only
true for the converged solution, but not for iterates. In this sense, we have
also proposed the concept of projected nonlinear solvers, where a projection
step is performed at the end of every nonlinear iteration onto a FE space of
admissible solutions. The space of admissible solutions is the one that
satisfies the desired monotonic properties (maximum principle or positivity).

14
The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
2
The categories of this paper are: math
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess