THE ERROR ESTIMATION IN FINITE ELEMENT METHODS FOR ELLIPTIC EQUATIONS WITH LOW REGULARITY
Jing Yang
10.25394/PGS.12249725.v1
https://hammer.figshare.com/articles/thesis/THE_ERROR_ESTIMATION_IN_FINITE_ELEMENT_METHODS_FOR_ELLIPTIC_EQUATIONS_WITH_LOW_REGULARITY/12249725
<div>
<div>
<div>
<p>This dissertation contains two parts: one part is about the error estimate for the
finite element approximation to elliptic PDEs with discontinuous Dirichlet boundary
data, the other is about the error estimate of the DG method for elliptic equations
with low regularity.
</p>
<p>Elliptic problems with low regularities arise in many applications, error estimate
for sufficiently smooth solutions have been thoroughly studied but few results have
been 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 are
not 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 continuous
approximation of the Dirichlet boundary data. With discontinuous boundary data,
the variational form is not valid since the solution for the general elliptic equations
is 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 general
elliptic equations.
</p>
<p>Part II presents a class of DG methods and proves the stability when the solution belong to H1+ε where ε < 1/2 could be very small. we derive a non-standard
variational formulation for advection-diffusion-reaction problems. The formulation is
defined in an appropriate function space that permits discontinuity across element
</p>
</div>
</div>
<div>
<div>
<p>viii
</p>
</div>
</div>
</div>
<div>
<div>
<div>
<p>interfaces and does not require piece wise Hs(Ω), s ≥ 3/2, smoothness. Hence, both
continuous and discontinuous (including Crouzeix-Raviart) finite element spaces may
be used and are conforming with respect to this variational formulation. Then it establishes the a priori error estimates of these methods when the underlying problem
is not piece wise H3/2 regular. The constant in the estimate is independent of the
parameters of the underlying problem. Error analysis presented here is new. The
analysis makes use of the discrete coercivity of the bilinear form, an error equation,
and an efficiency bound of the continuous finite element approximation obtained in
the 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 also
proves the stability.
</p>
</div>
</div>
</div>
2020-05-05 17:32:59
Finite Element Methods (FEM)
Error Estimates
a priori error analysis