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We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.

A cut cell based sharp-interface Runge–Kutta discontinuous Galerkin method, with quadtree-like adaptive mesh refinement, is developed for simulating compressible two-medium flows with clear interfaces. In this approach, the free interface is represented by curved cut faces and evolved by solving the level-set equation with high order upstream central scheme. Thus every mixed cell is divided into two cut cells by a cut face. The Runge–Kutta discontinuous Galerkin method is applied to calculate each single-medium flow governed by the Euler equations. A two-medium exact Riemann solver is applied on the cut faces and the Lax–Friedrichs flux is applied on the regular faces. Refining and coarsening of meshes occur according to criteria on distance from the material interface and on magnitudes of pressure/density gradient, and the solutions and fluxes between upper-level and lower-level meshes are synchronized by $$L^2$$ projections to keep conservation and high order accuracy. This proposed method inherits the advantages of the discontinuous Galerkin method (compact and high order) and cut cell method (sharp interface and curved cut face), thus it is fully conservative, consistent, and is very accurate on both interface and flow field calculations. Numerical tests with a variety of parameters illustrate the accuracy and robustness of the proposed method.

In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $$C^0$$ -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ and in $$L^\infty (H^1)$$ -norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Explicita posteriori residual type error estimators in L2(H1) norm are derived for discontinuous Galerkin (DG) methods applied to transport in porous media with general kinetic reactions. They are flexible and apply to all the four primal DG schemes, namely, Oden–Babuška–Baumann DG (OBB-DG), non-symmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG) and incomplete interior penalty Galerkin (IIPG). The error estimators use directly all the available information from the numerical solution and can be computed efficiently. Numerical examples are presented to demonstrate the efficiency and the effectivity of these theoretical estimators.

In Liu and Qiu (J Sci Comput 63:548–572, 2015), we presented a class of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes for conservation laws, in which the reconstruction of fluxes is based on the usual practice of reconstructing the flux functions. In this follow-up paper, we present an alternative formulation to reconstruct the numerical fluxes, in which we first use the solution and its derivatives directly to interpolate point values at interfaces of computational cells, then we put the point values at interface of cell in building block to generate numerical fluxes. The building block can be arbitrary monotone fluxes. Comparing with Liu and Qiu (2015), one major advantage is that arbitrary monotone fluxes can be used in this framework, while in Liu and Qiu (2015) the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. Furthermore, these new schemes still keep the effectively narrower stencil of HWENO schemes in the process of reconstruction. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to demonstrate the good performance of the methods.

The Swift–Hohenberg equation as a central nonlinear model in modern physics has a gradient flow structure. Here we introduce fully discrete discontinuous Galerkin (DG) schemes for a class of fourth order gradient flow problems, including the nonlinear Swift–Hohenberg equation, to produce free-energy-decaying discrete solutions, irrespective of the time step and the mesh size. We exploit and extend the mixed DG method introduced in Liu and Yin (J Sci Comput 77:467–501, 2018) for the spatial discretization, and the “Invariant Energy Quadratization” method for the time discretization. The resulting IEQ-DG algorithms are linear, thus they can be efficiently solved without resorting to any iteration method. We actually prove that these schemes are unconditionally energy stable. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and energy stability of our new algorithm. The numerical results on two dimensional pattern formation problems indicate that the method is able to deliver comparable patterns of high accuracy.

In this paper, we study the $$H^1$$ -conforming virtual element method for the spatial discretization of semilinear parabolic equations with inhomogeneous Dirichlet boundary conditions based on a modified symmetric Nitsche’s formula. Herein, the modified Nitsche’s formula is constructed by introducing a global lifting operator which maps the trace of an $$H^1$$ -function into the global conforming virtual element space. In contrast to the classical symmetric Nitsche’s method, the penalty parameter in the modified symmetric Nitsche’s method does not need to be greater than a strictly positive lower bound, and it only needs to be greater than 0 to provide a coercive spatial bilinear form. For time discretization, the second-order backward difference formula is used. On the basis of some assumptions on the given problem data, the optimal error estimates in both an energy norm and $$L^2$$ -norm are established for the semi-discrete and fully discrete schemes. The theoretical results are verified by some numerical experiments.

We propose.