Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

In this paper I present a history of tractability of continuous problems, which has its beginning in the successful numerical tests for highdimensional integration of finance problems. Tractability results will be illustrated for two multivariate problems, integration and linear tensor products problems, in the worst case setting. My talk at FoCM'08 in Hong Kong and this paper are based on the book Tractability of Multivariate Problems , written jointly with Erich Novak. The first volume of our book has been recently published by the European Mathematical Society. Introduction Many people have recently become interested in studying the tractability of continuous problems. This area of research addresses the computational complexity of multivariate problems defined on spaces of functions of d variables, with d that can be in the hundreds or thousands; in fact, d can even be arbitrarily large. Such problems occur in numerous applications including physics, chemistry, finance, economics, and the computational sciences. As with all problems arising in information-based complexity, we want to solve multivariate problems to within ∈, using algorithms that use finitely many functions values or values of some linear functionals. Let n (∈, d ) be the minimal number of function values or linear functionals that is needed to compute the solution of the d -variate problem to within ∈. For many multivariate problems defined over standard spaces of functions n (∈, d ) is exponentially large in d .

Tractability of Multivariate Problems

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

/pdf/quasi-optimal-convergence-rate-for-an-adaptive-finite-4lmcxnlu35.pdf

Quasi-Optimal Convergence Rate for an 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 e > 0 in energy norm by a continuous piecewise linear function on some partition with O(e-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.

Optimality of a Standard Adaptive Finite Element Method

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.

The completion of locally refined simplicial partitions created by bisection

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.

https://www.hcm.uni-bonn.de/fileadmin/user_upload/sparcity_and_computation/Stevenson.pdf

Space-time adaptive wavelet methods for parabolic evolution problems

In "Adaptive wavelet methods II---Beyond the elliptic case" of Cohen, Dahmen, and DeVore [Found. Comput. Math., 2 (2002), pp. 203--245], an adaptive method has been developed for solving general operator equations. Using a Riesz basis of wavelet type for the energy space, the operator equation is transformed into an equivalent matrix-vector system. This system is solved iteratively, where the application of the infinite stiffness matrix is replaced by an adaptive approximation. Assuming that the stiffness matrix is sufficiently compressible, i.e., that it can be sufficiently well approximated by sparse matrices, it was proved that the adaptive method has optimal computational complexity in the sense that it converges with the same rate as the best N-term approximation for the solution, assuming that the latter would be explicitly available. The condition concerning compressibility requires that, dependent on their order, the wavelets have sufficiently many vanishing moments, and that they be sufficiently smooth. However, except on tensor product domains, wavelets that satisfy this smoothness requirement are not easy to construct.
In this paper we write the domain or manifold on which the operator equation is posed as an overlapping union of subdomains, each of them being the image under a smooth parametrization of the hypercube. By lifting wavelets on the hypercube to the subdomains, we obtain a {\em frame} for the energy space. With this frame the operator equation is transformed into a matrix-vector system, after which this system is solved iteratively by an adaptive method similar to the one from the work of Cohen, Dahmen, and DeVore. With this approach, frame elements that have sufficiently many vanishing moments and are sufficiently smooth, something needed for the compressibility, are easily constructed. By handling additional difficulties due to the fact that a frame gives rise to an underdetermined matrix-vector system, we prove that this adaptive method has optimal computational complexity.

/pdf/adaptive-solution-of-operator-equations-using-wavelet-frames-1k9zq08pe9.pdf

Adaptive Solution of Operator Equations Using Wavelet Frames

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 $s$ 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.

/pdf/element-by-element-construction-of-wavelets-satisfying-2t7ol3fvpm.pdf

Rob Stevenson

Papers

Optimality of a Standard Adaptive Finite Element Method

The completion of locally refined simplicial partitions created by bisection

Space-time adaptive wavelet methods for parabolic evolution problems

Adaptive Solution of Operator Equations Using Wavelet Frames

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