Abstract
This is a survey on the theory of adaptive finite element methods
(AFEM), which are fundamental in modern computational science and
engineering. We present a self-contained and up-to-date discussion
of AFEM for linear second order elliptic partial differential equations
(PDEs) and dimension d>1, with emphasis on the differences and advantages
of AFEM over standard FEM. The material is organized in chapters
with problems that extend and complement the theory. We start with
the functional framework, inf-sup theory, and Petrov-Galerkin method,
which are the basis of FEM. We next address four topics of essence
in the theory of AFEM that cannot be found in one single article:
mesh refinement by bisection, piecewise polynomial approximation
in graded meshes, a posteriori error analysis, and convergence and
optimal decay rates of AFEM. The first topic is of geometric and
combinatorial nature, and describes bisection as a rather simple
and efficient technique to create conforming graded meshes with optimal
complexity. The second topic explores the potentials of FEM to compensate
singular behavior with local resolution and so reach optimal error
decay. This theory, although insightful, is insufficient to deal
with PDEs since it relies on knowing the exact solution. The third
topic provides the missing link, namely a posteriori error estimators,
which hinge exclusively on accessible data: we restrict ourselves
to the simplest residual-type estimators and present a complete discussion
of upper and lower bounds, along with the concept of oscillation
and its critical role. The fourth topic refers to the convergence
of adaptive loops and its comparison with quasi-uniform refinement.
We first show, under rather modest assumptions on the problem class
and AFEM, convergence in the natural norm associated to the variational
formulation. We next restrict the problem class to coercive symmetric
bilinear forms, and show that AFEM is a contraction for a suitable
error notion involving the induced energy norm. This property is
then instrumental to prove optimal cardinality of AFEM for a class
of singular functions, for which the standard FEM is suboptimal.
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