scholarly journals High Accuracy Numerical Solution of Elliptic Equations with Discontinuous Coefficients

2021 ◽  
Vol 14 (4) ◽  
pp. 88-101
Author(s):  
V.P. Shapeev ◽  
◽  
L.S. Bryndin ◽  
V.A. Belyaev ◽  
Keyword(s):  
Numerical Solution ◽  
Elliptic Equations ◽  
Discontinuous Coefficients ◽  
High Accuracy

Numerical Solution of the Two-Phase Stefan Problem in the Enthalpy Formulation with Smoothing the Coefficients

2021 ◽  
pp. 4-23
Author(s):  
V.I. Vasilyev ◽  
M.V. Vasilyeva ◽  
S.P. Stepanov ◽  
N.I. Sidnyaev ◽  
O.I. Matveeva ◽  
...  
Keyword(s):  
Numerical Solution ◽  
Stefan Problem ◽  
Large Diameter ◽  
Discontinuous Coefficients ◽  
Numerical Calculations ◽  
Smooth Functions ◽  
Two Phase ◽  
Time Layer ◽  
Dirac Delta ◽  
Selection Of

To simulate heat transfer processes with phase transitions, the classical enthalpy model of Stefan is used, accompanied by phase transformations of the medium with absorption and release of latent heat of a change in the state of aggregation. The paper introduces a solution to the two-phase Stefan problem for a one-dimensional quasilinear second-order parabolic equation with discontinuous coefficients. A method for smearing the Dirac delta function using the smoothing of discontinuous coefficients by smooth functions is proposed. The method is based on the use of the integral of errors and the Gaussian normal distribution with an automated selection of the value of the interval of their smoothing by the desired function (temperature). The discontinuous coefficients are replaced by bounded smooth temperature functions. For the numerical solution, the finite difference method and the finite element method with an automated selection of the smearing and smoothing parameters for the coefficients at each time layer are used. The results of numerical calculations are compared with the solution of Stefan’s two-phase self-similar problem --- with a mathematical model of the formation of the ice cover of the reservoir. Numerical simulation of the thawing effect of installing additional piles on the existing pile field is carried out. The temperature on the day surface of the base of the structure is set with account for the amplitude of air temperature fluctuations, taken from the data of the Yakutsk meteorological station. The study presents the results of numerical calculations for concrete piles installed in the summer in large-diameter drilled wells using cement-sand mortars with a temperature of 25 °С. The distributions of soil temperature are obtained for different points in time

Elliptic equations with discontinuous coefficients in generalized Morrey spaces

2017 ◽  
Vol 3 (3) ◽  
pp. 728-762 ◽  
Author(s):  
Giuseppe Di Fazio ◽  
Denny Ivanal Hakim ◽  
Yoshihiro Sawano
Keyword(s):  
Elliptic Equations ◽  
Discontinuous Coefficients ◽  
Morrey Spaces ◽  
Generalized Morrey Spaces

scholarly journals A BDDC Preconditioner for the RotatedQ1FEM for Elliptic Problems with Discontinuous Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yaqin Jiang
Keyword(s):  
Finite Element Method ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Mesh Size ◽  
Discontinuous Coefficients ◽  
Schwarz Methods ◽  
Additive Schwarz ◽  
Variational Form ◽  
Second Order Elliptic Equations ◽  
Additive Schwarz Methods

We propose a BDDC preconditioner for the rotatedQ1finite element method for second order elliptic equations with piecewise but discontinuous coefficients. In the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasioptimal convergence behavior; that is, the condition number of the preconditioned problem is independent of the jumps of the coefficients and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.

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