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In this work, we propose a nonlinear stabilization technique for scalarconservation laws with implicit time stepping. The method relies on anartificial diffusion method, based on a graph-Laplacian operator. It isnonlinear, since it depends on a shock detector. The same shock detector isused to gradually lump the mass matrix. The resulting method is LED, positivitypreserving, linearity preserving, and also satisfies a global DMP. Lipschitzcontinuity has also been proved. However, the resulting scheme is highlynonlinear, leading to very poor nonlinear convergence rates. We propose asmooth version of the scheme, which leads to twice differentiable nonlinearstabilization schemes. It allows one to straightforwardly use Newton's methodand obtain quadratic convergence. In the numerical experiments, steady andtransient linear transport, and transient Burgers' equation have beenconsidered in 2D. Using the Newton method with a smooth version of the schemewe can reduce 10 to 20 times the number of iterations of Anderson accelerationwith the original non-smooth scheme. In any case, these properties are onlytrue for the converged solution, but not for iterates. In this sense, we havealso proposed the concept of projected nonlinear solvers, where a projectionstep is performed at the end of every nonlinear iteration onto a FE space ofadmissible solutions. The space of admissible solutions is the one thatsatisfies the desired monotonic properties (maximum principle or positivity).

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization