The aim of this work is to design monotonicity-preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear (shock-capturing) stabilization. As linear stabilization, we consider and analyze a novel symmetric projection stabilization technique based on a local Scott-Zhang projector. Next, we design a weighting of the aforementioned linear stabilization such that, when combined with a finite element discretization enjoying a discrete maximum principle (usually attained via nonlinear stabilization), it does not spoil these monotonicity properties. Then, we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes. For the nodal scheme, we prove a discrete maximum principle for time-dependent multidimensional transport problems. Numerical experiments support the numerical analysis and we show that the resulting methods provide excellent results. In particular, we observe that the proposed nonlinear stabilization techniques do an excellent job eliminating oscillations around shocks.

This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.Peer ReviewedPostprint (author's final draft

The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection-diffusion problems. To do so, a novel definition of discrete maximum principle (DMP) is proposed using the discrete variational setting of the problem, and we show that the fulfilment of this DMP implies that the minimum/maximum (depending on the sign of the forcing term) is on the boundary for multidimensional problems. Then, an artificial viscosity (AV) technique is designed for convection-dominant problems that satisfies the above mentioned DMP. The noncomplete stabilized interior penalty dG method is proved to fulfil the DMP property for the one-dimensional linear case when adding such AV with certain parameters. The benchmarks for the constant values to satisfy the DMP are calculated and tested in the numerical experiments section. Finally, the method is applied to different test problems in one and two dimensions to show its performance.

The aim of this work is to propose an hp-adaptive algorithm for discontinuous Galerkin methods that is capable to detect the discontinuities and sharp layers and avoid the spurious oscillation of the solution around them. In order to control the spurious oscillations, artificial viscosity is used with the particularity that it is only applied around the layers where the solution changes abruptly. To do so, a novel troubled-cell detector has been developed in order to mark the elements around those layers and to impose linear order in them. The detector takes advantage of the evolution of the value of the gradient through the adaptive process.