In this thesis, we analyze an adaptive discontinuous finite element method for symmetric
second order linear elliptic operators. Moreover, we obtain a fully computable convergence
analysis on the broken energy seminorm in first order symmetric interior penalty discontin-
uous Galerkin finite element approximations of this problem. The method is formulated on
nonconforming meshes made of triangular elements with first order polynomial in two di-
mension. We use an estimator which is completely free of unknown constants and provide a
guaranteed numerical bound on the broken energy norm of the error. This estimator is also
shown to provide a lower bound for the broken energy seminorm of the error up to a constant
and higher order data oscillation terms. Consequently, the estimator yields fully reliable,
quantitative error control along with efficiency.
As a second problem, explicit expression for constants of the inverse inequality are given in
1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the
inclusion of mesh dependent terms in new classes of methods for a variety of applications.
Several inequalities of functional analysis are often employed in convergence proofs. Inverse
estimates have been used extensively in the analysis of finite element methods. It is char-
acterized as tools for the error analysis and practical design of finite element methods with
terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality
is provided in this thesis.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12614345/index.pdf |
| Date | 01 May 2012 |
| Creators | Ozisik, Sevtap |
| Contributors | Merdan, Songul |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | Ph.D. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for public access |
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