Hybrid high-order method for singularly perturbed fourth-order problems on curved domains

We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilisation operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.

Guaranteed lower bounds on eigenvalues of elliptic operators with a hybrid high-order method

This paper introduces a novel hybrid high-order (HHO) method to approximate the eigenvalues of a symmetric compact differential operator. The HHO method combines two gradient reconstruction operators by means of a parameter $$0<\alpha <~1$$ 0 < α < 1 and introduces a novel cell-based stabilization operator weighted by a parameter $$0<\beta <\infty $$ 0 < β < ∞ . Sufficient conditions on the parameters $$\alpha $$ α and $$\beta $$ β are identified leading to a guaranteed lower bound property for the discrete eigenvalues. Moreover optimal convergence rates are established. Numerical studies for the Dirichlet eigenvalue problem of the Laplacian provide evidence for the superiority of the new lower eigenvalue bounds compared to previously available bounds.

A Hybrid High-Order method for creeping flows of non-Newtonian fluids

In this paper, we design and analyze a Hybrid High-Order discretization method for the steady motion of non-Newtonian, incompressible fluids in the Stokes approximation of small velocities. The proposed method has several appealing features including the support of general meshes and high-order, unconditional inf-sup stability, and orders of convergence that match those obtained for Leray-Lions scalar problems. A complete well-posedness and convergence analysis of the method is carried out under new, general assumptions on the strain rate-shear stress law, which encompass several common examples such as the power-law and Carreau-Yasuda models. Numerical examples complete the exposition.

Studying and Analysis of a Novel RK-Sinc Scheme

In this paper, a novel high-order method, Runge-Kutta Sinc (RK-Sinc), is proposed. The RK-Sinc scheme employs the strong stability preserving Runge-Kutta (SSP-RK) algorithm to substitute time derivative and the Sinc function to replace spatial derivates. The computational efficiency, numerical dispersion and convergence of the RK-Sinc algorithm are addressed. The proposed method presents the better numerical dispersion and the faster convergence rate both in time and space domain. It is found that the computational memory of the RK-Sinc is more than two times of the FDTD for the same stencil size. Compared with the conventional FDTD, the new scheme provides more accuracy and great potential in computational electromagnetic field.