scholarly journals A priori error estimates of regularized elliptic problems

2020 ◽  
Vol 146 (3) ◽  
pp. 571-596
Author(s):  
Luca Heltai ◽  
Wenyu Lei
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
Elliptic Problems ◽  
Mesh Size ◽  
Rates Of Convergence ◽  
Smooth Functions ◽  
Approximation Process ◽  
Immersed Interface Method ◽  
Dirac Delta ◽  
Delta Distribution

Abstract Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

scholarly journals Optimal a priori error estimates in weighted Sobolev spaces for the Poisson problem with singular sources

2020 ◽  
Author(s):  
Ignacio Ojea
Keyword(s):  
A Priori ◽  
Weighted Sobolev Spaces ◽  
Rates Of Convergence ◽  
Three Dimensions ◽  
Galerkin Projection ◽  
The Finite Element Method ◽  
Dirac Delta ◽  
Poisson Problem ◽  
Graded Meshes ◽  
The Stability

We study the problem $-\Delta u = f$, where $f$ has a point-singularity. In particular, we are interested in $f=\delta_{x_0}$, a Dirac delta with support in $x_0$, but singularities of the form $f\sim |x-x_0|^{-s}$ are also considered. We prove the stability of the Galerkin projection on graded meshes in weighted spaces, with weights given by powers of the distance to $x_0$. We also recover optimal rates of convergence for the finite element method on these graded meshes. Our approach is general and holds both in two and three dimensions.  Numerical experiments are shown that verify our results, and lead to interesting observations.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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

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