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.

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 Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem

2015 ◽  
Vol 15 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Thirupathi Gudi ◽  
Kamana Porwal
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lagrange Multiplier ◽  
Obstacle Problem ◽  
Posteriori Error ◽  
Optimal Order ◽  
Error Estimator ◽  
A Posteriori ◽  
Discrete Solution ◽  
Element Method

AbstractA residual based a posteriori error estimator is derived for a quadratic finite element method (FEM) for the elliptic obstacle problem. The error estimator involves various residuals consisting of the data of the problem, discrete solution and a Lagrange multiplier related to the obstacle constraint. The choice of the discrete Lagrange multiplier yields an error estimator that is comparable with the error estimator in the case of linear FEM. Further, an a priori error estimate is derived to show that the discrete Lagrange multiplier converges at the same rate as that of the discrete solution of the obstacle problem. The numerical experiments of adaptive FEM show optimal order convergence. This demonstrates that the quadratic FEM for obstacle problem exhibits optimal performance.

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.

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