Optimal error estimates for the pressure stabilized Petrov–Galerkin (PSPG) method for the evolutionary Stokes equations are proved, in the case of regular solutions, without restriction on the length of the time step. These results clarify the “instability of the discrete pressure for small time steps” reported in the literature. First, the limit situation of the continuous-in-time discretizati...

We present a hybridized discontinuous Galerkin (HDG) solver for the time-dependent compressible Euler and Navier-Stokes equations. In contrast to discontinuous Galerkin (DG) methods, the number of globally coupled degrees of freedom is usually tremendously smaller for HDG methods, as these methods can rely on hybridization. However, applying the method to a time-dependent problem amounts to sol...

In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, b...

The time dependent Ginzburg-Landau (TDGL) equation is a typical model in phase field theory for many applications like two phase flow simulations and phase transitions. In this paper, we develop effective algorithms so that the solution of the TDGL model can be accurately approximated. Specifically, we adopt finite element methods for the spatial discretization and study different algorithms fo...

We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, finite volume, mixed finite element, and conforming and nonconforming finite element methods in space and the backward Euler scheme in time. Our estimates are based on a H-conforming reconstruction of the potential, continuous and piec...

This paper presents a sliding mode filter that effectively removes impulsive noise and high-frequency noise with producing much smaller phase lag than linear filters. It is less prone to overshoot than previous sliding mode filters and it does not produce chattering. In addition, it is computationally inexpensive, and thus suitable for realtime applications. The proposed sliding mode filter emp...

We formulate collocation Runge–Kutta time-stepping schemes applied to linear parabolic evolution equations as space-time Petrov–Galerkin discretizations, and investigate their a priori stability for the parabolic space-time norms, that is the continuity constant of the discrete solution mapping. We focus on collocation based on A-stable Gauss–Legendre and L-stable right-Radau nodes, addressing ...

The paper is concerned with the construction and convergence analysis of a discontinuous Galerkin finite element method for the Cahn–Hilliard equation with convection. Using discontinuous piecewise polynomials of degree p ≥ 1 and backward Euler discretization in time, we show that the order-parameter c is approximated in the broken L∞(H1) norm with optimal order, O(hp+ τ); the associated chemic...

We study the interaction between a poroelastic medium and a fracture filled with fluid. The flow in the fracture is described by the Brinkman equations for an incompressible fluid and the poroelastic medium by the quasi-static Biot model. The two models are fully coupled via the kinematic and dynamic conditions. The Brinkman equations are then averaged over the cross-sections, giving rise to a ...

Numerical methods for the primitive equations (PEs) of oceanic flow are presented in this paper. First, a two-dimensional Poisson equation with a suitable boundary condition is derived to solve the surface pressure. Consequently, we derive a new formulation of the PEs in which the surface pressure Poisson equation replaces the nonlocal incompressibility constraint, which is known to be inconven...