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.

Accuracy analysis of gradient reconstruction on isotropic unstructured meshes and its effects on inviscid flow simulation

The accuracy of gradient reconstruction methods on unstructured meshes is analyzed both mathematically and numerically. Mathematical derivations reveal that, for gradient reconstruction based on the Green-Gauss theorem (the GG methods), if the summation of first-and-lower-order terms does not counterbalance in the discretized integral process, which rarely occurs, second-order accurate approximation of face midpoint value is necessary to produce at least first-order accurate gradient. However, gradient reconstruction based on the least-squares approach (the LSQ methods) is at least first-order on arbitrary unstructured grids. Verifications are performed on typical isotropic grid stencils by analyzing the relationship between the discretization error of gradient reconstruction and the discretization error of the face midpoint value approximation of a given analytic function. Meanwhile, the numerical accuracy of gradient reconstruction methods is examined with grid convergence study on typical isotropic grids. Results verify the phenomenon of accuracy degradation for the GG methods when the face midpoint value condition is not satisfied. The LSQ methods are proved to be at least first-order on all tested isotropic grids. To study gradient accuracy effects on inviscid flow simulation, solution errors are quantified using the Method of Manufactured Solutions (MMS) which was validated before adoption by comparing with an exact solution case, i.e., the 2-dimensional (2D) inviscid isentropic vortex. Numerical results demonstrate that the order of accuracy (OOA) of gradient reconstruction is crucial in determining the OOA of numerical solutions. Solution accuracy deteriorates seriously if gradient reconstruction does not reach first-order.