scholarly journals Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

2012 ◽  
Vol 5 (1) ◽  
pp. 19-42 ◽  
Author(s):  
Armando Coco and Giovanni Russo
Keyword(s):  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Multigrid Methods ◽  
Second Order ◽  
One Dimensional

scholarly journals Explicit Estimates for Solutions of Mixed Elliptic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Luisa Consiglieri
Keyword(s):  
Potential Theory ◽  
Mixed Problem ◽  
Elliptic Problems ◽  
Discontinuous Coefficients ◽  
Divergence Form ◽  
Order Equation ◽  
Second Order ◽  
Green Functions ◽  
Mixed Problems ◽  
Quantitative Estimates

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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