Difference schemes for the dirichlet problem for an elliptical equation with discontinuous coefficients in an arbitrary region on a regular grid

1991 ◽  
Vol 54 (2) ◽  
pp. 747-751
Author(s):  
S. A. Voitsekhovskii ◽  
V. L. Makarov
Keyword(s):  
Dirichlet Problem ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Regular Grid ◽  
Elliptical Equation ◽  
Arbitrary Region

On Finite Difference Schemes for Elliptic Equations with Discontinuous Coefficients

2013 ◽  
Vol 13 (3) ◽  
pp. 281-289
Author(s):  
Manfred Dobrowolski
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elliptic Equations ◽  
Approximation Error ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Finite Element Method ◽  
Order Four ◽  
Element Theory

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

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