Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

摘要: 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 graph-Laplacian operator. It is nonlinear, since it depends shock detector. Further, the resulting linearity preserving. same detector used to gradually lump mass matrix. LED, positivity preserving, and also satisfies global DMP. Lipschitz continuity has been proved. However, scheme highly leading very poor convergence rates. We smooth version of scheme, which leads twice differentiable schemes. allows one straightforwardly use Newton’s obtain quadratic convergence. numerical experiments, steady transient linear transport, Burgers’ equation have considered in 2D. Using Newton can reduce 10 20 times number iterations Anderson acceleration original non-smooth scheme. any case, these properties are only true converged solution, but not iterates. sense, proposed concept projected solvers, where projection step performed at end every onto FE space admissible solutions. solutions that desired monotonic (maximum principle or positivity).