Residual-type a posteriori error estimator for a quasi-static Signorini contact problem

2019 ◽  
Vol 40 (3) ◽  
pp. 1937-1971 ◽  
Author(s):  
Mirjam Walloth
Keyword(s):  
Lower Bounds ◽  
Mesh Size ◽  
Type A ◽  
Contact Forces ◽  
Upper And Lower Bounds ◽  
Gap Functions ◽  
Error Estimator ◽  
A Posteriori ◽  
Time Step ◽  
Spatial Error

Abstract We present a new residual-type a posteriori estimator for a quasi-static Signorini problem. The theoretical results are derived for two- and three-dimensional domains and the case of nondiscrete gap functions is addressed. We derive global upper and lower bounds with respect to an error notion, which measures the error in the displacements, the velocities and a suitable approximation of the contact forces. Further, local lower bounds for the spatial error at each discrete time point are given. The estimator splits in temporal and spatial contributions, which can be used for the adaptation of the time step as well as the mesh size. In the derivation of the estimator the local properties of the solution are exploited such that the spatial estimator has no contributions related to the nonlinearities in the interior of the actual time-dependent contact zone, but gives rise to an appropriate refinement of the free boundary zone.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

2017 ◽  
Vol 7 (3) ◽  
pp. 508-529 ◽  
Author(s):  
Xiaobo Zheng ◽  
Xiaoping Xie
Keyword(s):  
Finite Element ◽  
Lower Bounds ◽  
Stokes Problem ◽  
Three Dimensions ◽  
Upper And Lower Bounds ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Velocity Approximation

AbstractA robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

A Posteriori Error Analysis for the Weak Galerkin Method for Solving Elliptic Problems

2018 ◽  
Vol 15 (08) ◽  
pp. 1850075 ◽  
Author(s):  
Tie Zhang ◽  
Yanli Chen
Keyword(s):  
Elliptic Problems ◽  
Posteriori Error ◽  
Upper And Lower Bounds ◽  
Posteriori Error Estimate ◽  
Error Estimator ◽  
A Posteriori ◽  
Weak Galerkin ◽  
A Posteriori Error ◽  
Galerkin Finite Element ◽  
Posteriori Error Analysis

In this paper, we study the a posteriori error estimate for weak Galerkin finite element method solving elliptic problems. A residual type error estimator is proposed and is proven to be reliable and efficient. This estimator provides global upper and lower bounds on the exact error in a discrete [Formula: see text]-norm. Numerical experiments are given to illustrate the effectiveness of the proposed error estimator.

scholarly journals An easily computable error estimator in space and time for the wave equation

2019 ◽  
Vol 53 (3) ◽  
pp. 729-747 ◽  
Author(s):  
O. Gorynina ◽  
A. Lozinski ◽  
M. Picasso
Keyword(s):  
Wave Equation ◽  
Error Estimator ◽  
A Posteriori ◽  
Discrete Solution ◽  
The Finite Element Method ◽  
Smooth Solutions ◽  
Time Step ◽  
Newmark Scheme ◽  
Second Order Wave Equation ◽  
Numer Anal

We propose a cheaper version of a posteriori error estimator from Gorynina et al. (Numer. Anal. (2017)) for the linear second-order wave equation discretized by the Newmark scheme in time and by the finite element method in space. The new estimator preserves all the properties of the previous one (reliability, optimality on smooth solutions and quasi-uniform meshes) but no longer requires an extra computation of the Laplacian of the discrete solution on each time step.

A POSTERIORI ERROR ESTIMATORS FOR MIXED APPROXIMATIONS OF EIGENVALUE PROBLEMS

1999 ◽  
Vol 09 (08) ◽  
pp. 1165-1178 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
LUCIA GASTALDI ◽  
CLAUDIO PADRA
Keyword(s):  
Eigenvalue Problems ◽  
Mixed Finite Element ◽  
Mesh Size ◽  
Mixed Finite Element Method ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Minimum Angle Condition ◽  
Posteriori Error Estimators

In this paper we introduce and analyze an a posteriori error estimator for the approximation of the eigenvalues and eigenvectors of a second-order elliptic problem obtained by the mixed finite element method of Raviart–Thomas of the lowest order. We define an error estimator of the residual type which can be computed locally from the approximate eigenvector and prove that the estimator is equivalent to the norm of the error in the approximation of the eigenvector up to higher order terms. The constants involved in this equivalence depend on the corresponding eigenvalue but are independent of the mesh size, provided the meshes satisfy the usual minimum angle condition. Moreover, the square root of the error in the approximation of the eigenvalue is also bounded by a constant times the estimator.

scholarly journals Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation

2021 ◽  
Author(s):  
Martyna Soszyńska ◽  
Thomas Richter
Keyword(s):  
Posteriori Error ◽  
Space Time ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimator ◽  
Time Step ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Time Stepping ◽  
Adaptive Time

AbstractWe study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime. Therefore we present two decoupling methods, namely a partitioned relaxation scheme and a shooting method. Furthermore, we develop an a posteriori error estimator serving as a mean for an adaptive time-stepping procedure. The goal is to optimally balance the time-step sizes of the two subproblems. The error estimator is based on the dual weighted residual method and relies on the space-time Galerkin formulation of the coupled problem. As an example, we take a linear set-up with the heat equation coupled to the wave equation. We formulate the problem in a monolithic manner using the space-time framework. In numerical test cases, we demonstrate the efficiency of the solution process and we also validate the accuracy of the a posteriori error estimator and its use for controlling the time-step sizes.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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