Rob Stevenson

Optimality of a Standard Adaptive Finite Element Method

In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance ε > 0 in energy norm by a continuous piecewise linear function on some partition with O(ε-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466-488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.

The completion of locally refined simplicial partitions created by bisection

Recently, in [Found. Comput. Math., 7(2) (2007), 245-269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466-488] by Morin, Nochetto, and Siebert, converges with the optimal rate. The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes. A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219-268] by Binev, Dahmen and DeVore saying that N - N-0 <= CM for some absolute constant C, where N-0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.

Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions

In this paper, we construct a class of locally supported wavelet bases for C0 Lagrange finite element spaces on possibly nonuniform meshes on \(n\)-dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces \(\mathcal{H}^s\) for \(|s|<\frac{3}{2}\) (\(|s|\leq 1\) on Lipschitz manifolds), and the wavelets can, in principle, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used, e.g., for constructing an optimal solver of discretized \(\mathcal{H}^s\)-elliptic problems for

Space-time adaptive wavelet methods for parabolic evolution problems

s

An optimal adaptive wavelet method without coarsening of the iterands

in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet bases can be implemented efficiently.

On the Compressibility of Operators in Wavelet Coordinates

With respect to space-time tensor-product wavelet bases, parabolic initial boundary value problems are equivalently formulated as bi-infinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the so-called curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension.

An Optimal Adaptive Finite Element Method

In this paper, an adaptive wavelet method for solving linear operator equations is constructed that is a modification of the method from [Math. Comp, 70 (2001), pp. 27-75] by Cohen, Dahmen and DeVore, in the sense that there is no recurrent coarsening of the iterands. Despite this, it will be shown that the method has optimal computational complexity. Numerical results for a simple model problem indicate that the new method is more efficient than an existing alternative adaptive wavelet method.

Piecewise Linear (Pre-)wavelets on Non-uniform Meshes

In [Found. Comput. Math., 2 (2002), pp. 203--245], Cohen, Dahmen, and DeVore proposed an adaptive wavelet algorithm for solving operator equations. Assuming that the operator defines a boundedly in...

Adaptive wavelet algorithms for elliptic PDE's on product domains

Although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than nonadaptive ones, usually they are not even proven to converge. Only recently in the work of Dorfler [SIAM J. Numer. Anal., 33 (1996), pp. 1106--1124] and that of Morin, Nochetto, and Siebert [SIAM J. Numer. Anal., 38 (2000), pp. 466--488], adaptive methods were constructed for which convergence could be demonstrated. However, convergence alone does not imply that the method is more efficient than its nonadaptive counterpart. In [ Numer. Math.}, 97 (2004), pp. 219--268], Binev, Dahmen, and DeVore added a coarsening step to the routine of Morin, Nochetto, and Siebert, and proved that the resulting method is quasi-optimal in the following sense: If the solution is such that for some s > 0, the error in energy norm of the best continuous piecewise linear approximations subordinate to any partition with n triangles is

Adaptive Wavelet Schemes for Parabolic Problems: Sparse Matrices and Numerical Results

\mathcal{O}(n^{-s})