Sônia M. Gomes

An adaptive multiresolution scheme with local time stepping for evolutionary PDEs

We present a fully adaptive numerical scheme for evolutionary PDEs in Cartesian geometry based on a second-order finite volume discretization. A multiresolution strategy allows local grid refinement while controlling the approximation error in space. For time discretization we use an explicit Runge-Kutta scheme of second-order with a scale-dependent time step. On the finest scale the size of the time step is imposed by the stability condition of the explicit scheme. On larger scales, the time step can be increased without violating the stability requirement of the explicit scheme. The implementation uses a dynamic tree data structure. Numerical validations for test problems in one space dimension demonstrate the efficiency and accuracy of the local time-stepping scheme with respect to both multiresolution scheme with global time stepping and finite volume scheme on a regular grid. Fully adaptive three-dimensional computations for reaction-diffusion equations illustrate the additional speed-up of the local time stepping for a thermo-diffusive flame instability.

A Fully Adaptive Multiresolution Scheme for Shock Computations

The scheme is based on Ami Harten’s ideas (Harten , 1994), the main tools coming from wavelet theory, in the framework of multiresolution analysis for cell averages. But instead of evolving cell averages on the finest uniform level, we propose to evolve just the cell averages on the grid determined by the significant wavelet coefficients. Typically, there are few cells in each time step, big cells on smooth regions, and smaller ones close to irregularities of the solution. For the numerical flux, we use a simple uniform central finite difference scheme, adapted to the size of each cell. If any of the required neighboring cell averages is not present, it is interpolated from coarser scales. But we switch to ENO scheme in the finest part of the grids. To show the feasibility and efficiency of the method, it is applied to a system arising in polymer-flooding of an oil reservoir. In terms of CPU time and memory requirements, it outperforms Harten’s multiresoltution algorithm.

A finite element model for three dimensional hydraulic fracturing

This paper is devoted to the development of a model for the numerical simulation of hydraulic fracturing processes with 3d fracture propagation. It takes into account the effects of fluid flow inside the fracture, fluid leak-off through fracture walls and elastic response of the surrounding porous media. Finite element techniques are adopted for the discretization of the conservation law of fluid flow and the singular integral equation relating the traction and the fracture opening. The discrete model for the singular integral equation is implemented using a stencil matrix structure allowing an efficient implementation of the fluid-structure interaction problem. Newtons method combined with GMRES linear solver are used to solve the resulting nonlinear set of equations. An algorithm for fracture propagation is proposed which is based on the balance of the amount of fluid transported to a certain point with the amount of fluid that could be lost by leak-off. To illustrate the feasibility of the model, we present simulation results for typical operational parameters.

Convergence estimates for the wavelet Galerkin method

This paper presents an analysis of the Galerkin approximation of a time dependent initial value problem correctly posed in the Petrovskii sense. The approximating spaces

Some results on the convergence of sampling series based on convolution integrals

\mathcal {V}_h

A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces

are spanned by translations and dilations of a single function

Wavelets and adaptive grids for the discontinuous Galerkin method

\Phi

Interpolating Wavelets and Adaptive Finite Difference Schemes for Solving Maxwell's Equations: The Effects of Gridding

, and Fourier techniques are used to analyze the accuracy of the method. This kind of procedure has already been applied in the literature for spline approximations. Our purpose here is to point out that the same methodology can be used for wavelet-based methods since the hypotheses required are automatically satisfied in the context of wavelet analysis. For instance, the basic function

Adaptive wavelet representation and differentiation on block-structured grids

\Phi

Convergence estimates for the wavelet-Galerkin method: superconvergence at the node points

is supposed to be regular, which means that