scholarly journals A priori error analysis for a finite element approximation of dynamic viscoelasticity problems involving a fractional order integro-differential constitutive law

2021 ◽  
Vol 47 (3) ◽  
Author(s):  
Yongseok Jang ◽  
Simon Shaw
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Fractional Order ◽  
A Priori ◽  
Finite Element Approximation ◽  
Constitutive Law ◽  
Element Approximation ◽  
Dynamic Viscoelasticity ◽  
A Priori Error Analysis

A Priori Error Analysis for the hp-Version of the Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

2003 ◽  
Vol 3 (4) ◽  
pp. 596-607 ◽  
Author(s):  
Igor Mozolevski ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Biharmonic Equation ◽  
A Priori ◽  
Element Approximation ◽  
Main Concern ◽  
Galerkin Finite Element ◽  
A Priori Error Analysis ◽  
Discontinuous Galerkin Finite Element

AbstractWe consider the hp-version of the discontinuous Galerkin finite element approximation of boundary value problems for the biharmonic equation. Our main concern is the a priori error analysis of the method, based on a nonsymmetric bilinear form with interior discontinuity penalization terms. We establish an a priori error bound for the method which is of optimal order with respect to the mesh size h , and nearly optimal with respect to the degree p of the polynomial approximation. For analytic solutions, the method exhibits an exponential rate of convergence under p- refinement. These results are shown in the DG-norm for a general shape regular family of partitions consisting of d-dimensional parallelepipeds. The theoretical results are confirmed by numerical experiments. The method has also been tested on several practical problems of thin-plate-bending theory and has been shown to be competitive in accuracy with existing algorithms.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

scholarly journals AN IMPROVED FINITE ELEMENT APPROXIMATION AND SUPERCONVERGENCE FOR TEMPERATURE CONTROL PROBLEMS

2013 ◽  
Vol 18 (5) ◽  
pp. 631-640 ◽  
Author(s):  
Yuelong Tang
Keyword(s):  
Finite Element ◽  
Temperature Control ◽  
Piecewise Linear ◽  
A Priori ◽  
Finite Element Approximation ◽  
Linear Functions ◽  
Control Problems ◽  
Element Approximation ◽  
Adjoint State ◽  
Admissible Controls

In this paper, we consider an improved finite element approximation for temperature control problems, where the state and the adjoint state are discretized by piecewise linear functions while the control is not discretized directly. The numerical solution of the control is obtained by a projection of the adjoint state to the set of admissible controls. We derive a priori error estimates and superconvergence of second-order. Moreover, we present some numerical examples to illustrate our theoretical results.

scholarly journals AN H1 -GALERKIN MIXED FINITE ELEMENT APPROXIMATION OF A NONLOCAL HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (5) ◽  
pp. 643-653
Author(s):  
Fengxin Chen ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Hyperbolic Equation ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Nonlinear Hyperbolic Equation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

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