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- W3023731497 abstract "This dissertation contains two parts: one part is about the error estimate for thefinite element approximation to elliptic PDEs with discontinuous Dirichlet boundarydata, the other is about the error estimate of the DG method for elliptic equationswith low regularity.Elliptic problems with low regularities arise in many applications, error estimatefor sufficiently smooth solutions have been thoroughly studied but few results havebeen obtained for elliptic problems with low regularities. Part I provides an error estimate for finite element approximation to elliptic partial differential equations (PDEs)with discontinuous Dirichlet boundary data. Solutions of problems of this type arenot in H1 and, hence, the standard variational formulation is not valid. To circumvent this difficulty, an error estimate of a finite element approximation in the W1,r(Ω)(0 < r < 2) norm is obtained through a regularization by constructing a continuousapproximation of the Dirichlet boundary data. With discontinuous boundary data,the variational form is not valid since the solution for the general elliptic equationsis not in H1. By using the W1,r (1 < r < 2) regularity and constructing continuous approximation to the boundary data, here we present error estimates for generalelliptic equations.Part II presents a class of DG methods and proves the stability when the solution belong to H1+e where e < 1/2 could be very small. we derive a non-standardvariational formulation for advection-diffusion-reaction problems. The formulation isdefined in an appropriate function space that permits discontinuity across elementviiiinterfaces and does not require piece wise Hs(Ω), s ≥ 3/2, smoothness. Hence, bothcontinuous and discontinuous (including Crouzeix-Raviart) finite element spaces maybe used and are conforming with respect to this variational formulation. Then it establishes the a priori error estimates of these methods when the underlying problemis not piece wise H3/2 regular. The constant in the estimate is independent of theparameters of the underlying problem. Error analysis presented here is new. Theanalysis makes use of the discrete coercivity of the bilinear form, an error equation,and an efficiency bound of the continuous finite element approximation obtained inthe a posteriori error estimation. Finally a new DG method is introduced i to over-come the difficulty in convergence analysis in the standard DG methods and alsoproves the stability." @default.
- W3023731497 created "2020-05-13" @default.
- W3023731497 creator A5022801075 @default.
- W3023731497 date "2020-05-05" @default.
- W3023731497 modified "2023-09-23" @default.
- W3023731497 title "THE ERROR ESTIMATION IN FINITE ELEMENT METHODS FOR ELLIPTIC EQUATIONS WITH LOW REGULARITY" @default.
- W3023731497 doi "https://doi.org/10.25394/pgs.12249725.v1" @default.
- W3023731497 hasPublicationYear "2020" @default.
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