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

2021 ◽  
Author(s):  
Carsten Carstensen ◽  
Alexandre Ern ◽  
Sophie Puttkammer
Keyword(s):  
Convergence Rates ◽  
Sufficient Conditions ◽  
High Order ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Bounds ◽  
High Order Method ◽  
Numerical Studies ◽  
Gradient Reconstruction ◽  
Optimal Convergence Rates ◽  
Dirichlet Eigenvalue Problem

AbstractThis 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.

scholarly journals An unfitted hybrid high-order method for the Stokes interface problem

2020 ◽  
Author(s):  
Erik Burman ◽  
Guillaume Delay ◽  
Alexandre Ern
Keyword(s):  
Numerical Simulations ◽  
Convergence Rates ◽  
A Priori ◽  
High Order ◽  
Interface Problem ◽  
Order Method ◽  
High Order Method ◽  
Optimal Convergence Rates ◽  
A Cell ◽  
Priori Error Estimates

Abstract We design and analyze a hybrid high-order method on unfitted meshes to approximate the Stokes interface problem. The interface can cut through the mesh cells in a very general fashion. A cell-agglomeration procedure prevents the appearance of small cut cells. Our main results are inf-sup stability and a priori error estimates with optimal convergence rates in the energy norm. Numerical simulations corroborate these results.

scholarly journals A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

2018 ◽  
Vol 18 (3) ◽  
pp. 467 ◽  
Author(s):  
Antônio José Boness Santos ◽  
C O Faria ◽  
Abimael F D Loula
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Rates Of Convergence ◽  
Processing Technique ◽  
Elasticity Problem ◽  
Hybrid Finite Element Method ◽  
Hybrid Finite Element ◽  
Numerical Studies ◽  
Optimal Convergence Rates

In this work, a primal hybrid finite element method for nearly incom pressible linear elasticity problem on triangular meshes is shown. This method consists of coupling local discontinuous Galerkin problems to the primal variable with a global problem for the Lagrange multiplier, which is identified as the trace of the displacement field. Also, a local post-processing technique is used to recover stress approximations with improved rates of convergence in H(div) norm. Numerical studies show that the method is locking free even using equal or different orders for displacement and stress fields and optimal convergence rates are obtained.

scholarly journals Eigenvalue bounds of mixed Steklov problems

2019 ◽  
Vol 22 (02) ◽  
pp. 1950008 ◽  
Author(s):  
Asma Hassannezhad ◽  
Ari Laptev
Keyword(s):  
Eigenvalue Problem ◽  
Fractional Laplacian ◽  
Eigenvalue Problems ◽  
Asymptotic Results ◽  
Riesz Means ◽  
Dirichlet Eigenvalue ◽  
Eigenvalue Bounds ◽  
Steklov Eigenvalue ◽  
Dirichlet Eigenvalue Problem ◽  
Neumann Eigenvalue

We study bounds on the Riesz means of the mixed Steklov–Neumann and Steklov–Dirichlet eigenvalue problem on a bounded domain [Formula: see text] in [Formula: see text]. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

The M/M/1 queue with mass exodus and mass arrivals when empty

1997 ◽  
Vol 34 (01) ◽  
pp. 192-207 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Sufficient Conditions ◽  
High Order ◽  
Necessary And Sufficient Conditions ◽  
Positive Recurrence ◽  
Powerful Technique ◽  
Resolvent Approach ◽  
Necessary And Sufficient ◽  
Direct Attack

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

Theoretical Aspects of a Multidomain High-Order Method for CAA

2003 ◽  
Author(s):  
Ronan Guenanff ◽  
Pierre Sagaut ◽  
Eric Manoha ◽  
Marc Terracol ◽  
Roger Lewandowsky
Keyword(s):  
High Order ◽  
Order Method ◽  
High Order Method

scholarly journals A Hybrid High-Order method for Kirchhoff–Love plate bending problems

2018 ◽  
Vol 52 (2) ◽  
pp. 393-421 ◽  
Author(s):  
Francesco Bonaldi ◽  
Daniele A. Di Pietro ◽  
Giuseppe Geymonat ◽  
Françoise Krasucki
Keyword(s):  
Sharp Estimate ◽  
High Order ◽  
Plate Bending ◽  
Mechanical Modeling ◽  
High Order Method ◽  
Local Polynomial ◽  
Bending Behavior ◽  
Bending Problems ◽  
Norm Of The Error ◽  
Theoretical Results

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in hk+3 is also derived for the L2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

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