Error estimates for the numerical solution of elliptic differential equations

1960 ◽  
Vol 5 (1) ◽  
pp. 293-306 ◽  
Author(s):  
Joachim Nitsche ◽  
Johannes C. C. Nitsche
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Error Estimates ◽  
Elliptic Differential Equations

scholarly journals Numerical Solution of Weakly Singular Integrodifferential Equations on Closed Smooth Contour in Lebesgue Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Feras M. Al Faqih
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Error Estimates ◽  
Closed Contour ◽  
Lebesgue Spaces ◽  
Solvability Conditions ◽  
Mechanical Quadrature ◽  
Weakly Singular ◽  
Quadrature Methods ◽  
Singular Integrodifferential Equations

The present paper deals with the justification of solvability conditions and properties of solutions for weakly singular integro-differential equations by collocation and mechanical quadrature methods. The equations are defined on an arbitrary smooth closed contour of the complex plane. Error estimates and convergence for the investigated methods are established in Lebesgue spaces.

The numerical solution of elliptic differential equations when the boundary conditions involve a derivative

1950 ◽  
Vol 242 (849) ◽  
pp. 345-378 ◽  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Elastic Plates ◽  
Boundary Values ◽  
Curved Boundaries ◽  
Elliptic Differential Equations ◽  
Fourth Order Equations ◽  
Second Order Equations ◽  
Approximate Equations ◽  
Laborious Method

The numerical solution of differential equations involves the replacement of derivatives by finitedifference equivalents. The idea of using approximate equations and subsequently correcting for the higher differences, already applied to second-order equations with specified boundary values, is here extended to the case where conditions at the boundary involve a derivative. The method is applied with examples to second- and fourth-order equations. The more difficult problems associated with curved boundaries are discussed, with particular reference to problems of stretching of flat elastic plates. An alternative but more laborious method of obtaining accurate solutions, the method of ‘the deferred approach to the lim it’, is illustrated by examples.

Error Estimates of Homogenization Theory

2012 ◽  
Vol 13 (3-4) ◽  
Author(s):  
Wei-Dong Song ◽  
Jian-Guo Ning ◽  
Jing Wang ◽  
Jian-Qiao Li
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Small Parameter ◽  
Error Estimates ◽  
Homogenization Theory ◽  
Basic Assumptions ◽  
Elliptic Differential Equations

AbstractBasic assumptions are introduced into a homogenization theory which is described in the form of elliptic differential equations with a small parameter. Homogenization operators are used to solve the equations by asymptotic expansion methods, and error estimates of homogenization solutions are given.

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