CONVERGENCE ANALYSIS OF THE PSEUDOSPECTRAL METHOD FOR LINEAR DAEs OF INDEX-2

2013 ◽  
Vol 10 (04) ◽  
pp. 1350019 ◽  
Author(s):  
F. GHANBARI ◽  
F. GHOREISHI
Keyword(s):  
Approximate Solution ◽  
Chebyshev Polynomials ◽  
Numerical Experiments ◽  
Pseudospectral Method ◽  
Posteriori Error ◽  
Differential Algebraic Equations ◽  
A Posteriori ◽  
Algebraic Equations ◽  
A Posteriori Error ◽  
Index 2

This paper is concerned with the study of pseudospectral discretizations of the differential-algebraic equations (DAEs). Pseudospectral method based on Chebyshev polynomials are used to transcribe a given DAE into a system of algebraic equations. A posteriori error bound between the desired solution to the index 2, DAEs and the pseudospectral approximate solution of the problem is estimated in the weighted L2 norm. Some numerical experiments are considered to demonstrate the efficiency and the applicability of the method.

scholarly journals Solution of Time-Varying Index-2 Linear Differential Algebraic Control Systems Via A Variational Formulation Technique

2021 ◽  
pp. 3656-3671
Author(s):  
Ghazwa F. Abd ◽  
Radhi A. Zaboon
Keyword(s):  
Approximate Solution ◽  
Variational Formulation ◽  
Direct Method ◽  
Differential Algebraic Equations ◽  
Time Varying ◽  
Algebraic Equations ◽  
Good Efficiency ◽  
Physical Problems ◽  
Index 2 ◽  
Linear Differential

    This paper deals with finding an approximate solution to the index-2 time-varying linear differential algebraic control system based on the theory of variational formulation. The solution of index-2 time-varying differential algebraic equations (DAEs) is the critical point of the equivalent variational formulation. In addition, the variational problem is transformed from the indirect into direct method by using a generalized Ritz bases approach. The approximate solution is found by solving an explicit linear algebraic equation, which makes the proposed technique reliable and efficient for many physical problems. From the numerical results, it can be implied that very good efficiency, accuracy, and simplicity of the present approach are obtained.

scholarly journals METHOD OF DETERMINING A POSTERIORI ERROR ESTIMATION IN CALCULATIONS OF COMPOSITE SHELLS USING MULTIGRID FINITE ELEMENTS

2018 ◽  
pp. 16-25
Author(s):  
Aleksandr Nikolaevich Grishanov
Keyword(s):  
Finite Elements ◽  
Error Estimates ◽  
Cylindrical Shells ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Suggested Method ◽  
Discrete Models ◽  
A Posteriori ◽  
Algebraic Equations ◽  
A Posteriori Error

A numerical method of determining a posteriori error estimates of solutions created for composite cylindrical shells using multigrid finite elements (MFE) has been proposed. The suggested method is based on ZZ method proposed by O. C. Zienkiewicz and J. Z. Zhu for both energy norm and L 2 norm of solution errors estimates. In contrast to ZZ method, the suggested method uses MFE that takes into account complex shapes, heterogeneous and micro-heterogeneous body structures and forms small dimension discrete models for creating «precise» solutions. To give examples there was carried out analysis of error estimates for displacements and stresses in calculation of stress-strain state (SSS) of three-layer cylindrical shells with and without cutouts under local loading. It has been stated that analysis of SSS using MFE causes converging sequences of approximate solutions in norm L 2. Calculations that use mean square error for stresses in each finite element of the shell show that MFE allow to use arbitrarily small regular discretization grids all over the shell area without the necessity to tighten the grid in local areas for calculating SSS. This leads to simple algorithms of calculating SSS with the help of MFE and ensures considerable saving of computer resources. In the given examples the use of MFE decreases the dimension of the system of MFE algebraic equations and reduces computer memory volume by 1 500 and 8∙104 times respectively, compared to the finite elements base model that doesn’t use MFE.

scholarly journals Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

2020 ◽  
Author(s):  
Svetoslav Nakov ◽  
Ekaterina Sobakinskaya ◽  
Thomas Renger ◽  
Johannes Kraus
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Electrostatic Coupling

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules is developed and verified. The development is based on a derivation of a goal-oriented a posteriori error estimates for the electrostatic coupling. These estimates involve the consideration of the primal and adjoint problems for the electrostatic potential of the system. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver<br>

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