Adaptive Finite Element Methods

TL;DR

This work presents a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error and proves linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes.

Abstract

This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second order elliptic PDEs and dimension d>1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs and beyond coercive problems to inf-sup stable AFEMs.

Figures (26)

• Figure 5.1: (Left) Piecewise linear basis function $\phi_z$ corresponding to interior node $z$; (Right) Support $\omega_z$ of $\phi_z$ and skeleton $\gamma_z$ (in solid line)
• Figure 5.2: Triangle $T\in\grid$ with vertex $v(T)$ and opposite refinement edge $E(T)$. The bisection rule for $d=2$ consists of connecting $v(T)$ with the midpoint of $E(T)$, thereby giving rise to children $T_1,T_2$ with common vertex $v(T_1)=v(T_2)$, the newly created vertex, and opposite refinement edges $E(T_1), E(T_2)$.
• Figure 5.3: Refinement of a single tetrahedron $\elm$ of type $t$. The child $\elm_1$ in the middle has the same node ordering regardless of type. In contrast, for the child $\elm_2$ on the right a triple is appended to two nodes. The local vertex index is given for these nodes by the $t$-th component of the triple.
• Figure 5.4: Sequence of bisection meshes $\{\gridk[k]\}_{k=0}^2$ starting from the initial mesh $\gridk[0]=\{T_i\}_{i=1}^4$ with longest edges labeled for bisection. Mesh $\gridk[1]$ is created from $\gridk[0]$ upon bisecting $T_1$ and $T_4$, whereas mesh $\gridk[2]$ arises from $\gridk[1]$ upon refining $T_6$ and $T_7$. The bisection rule is described in Figure \ref{['nv-F:edge-vertex']}.
• Figure 5.5: Forest $\forest_2$ corresponding to the grid sequence $\{\gridk[k]\}_{k=0}^2$ of Figure \ref{['nv-F:sequence']}. The roots of $\forest_2$ form the initial mesh $\gridk[0]$ and the leaves of $\forest_2$ constitute the conforming bisection mesh $\gridk[2]$. Moreover, each level of $\forest_2$ corresponds to all elements with generation equal to the level.

Theorems & Definitions (410)

• remark 1: equidistribution
• remark 2: nonlinear Approximation
• lemma 1: discrete inf-sup condition
• proposition 1: quasi-best-approximation property
• proof
• corollary 1: quasi-monotonicity
• proof
• proposition 2: quasi-interpolant without boundary values
• proposition 3: quasi-interpolant with boundary values
• remark 3: fractional regularity