A smooth fictitious domain/multiresolution method for elliptic equations on general domains

International audienceWe propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of corresponding discrete problem and the interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a cassini oval

Adaptive Wavelet Methods for Variational Formulations of Nonlinear Elliptic PDES on Tensor-Product Domains

This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by semilinear elliptic partial differential equations (PDEs). Semilinearity here refers to a special case of nonlinearity, i.e., the case of a linear operator combined with a nonlinear operator acting as a perturbation. In general, such BVPs are solved in an iterative fashion. It is, therefore, of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. Unlike the typical finite element method (FEM) theory for the numerical solution of the nonlinear operators, the new adaptive wavelet theory proposed in [Cohen.Dahmen.DeVore:2003:a, Cohen.Dahmen.DeVore:2003:b] guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets are the ideal candidate for this purpose since they allow to represent functions in infinite-dimensional general Banach or Hilbert spaces and operators on these. The purpose of adaptivity in the solution process of nonlinear PDEs is to invest extra degrees of freedom (DOFs) only where necessary, i.e., where the exact solution requires a higher number of function coefficients to represent it accurately. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the l_2 sequence spaces of expansion coefficients exist. This new paradigm presents nevertheless some problems in the design of practical algorithms. Imposing a certain structure, a tree structure, remedies these problems completely while restricting the applicability of the theoretical scheme only very slightly. It turns out that the considered approach naturally fits the theoretical background of nonlinear PDEs. The practical realization on a computer, however, requires to reduce the relevant ingredients to finite-dimensional quantities. It is this particular aspect that is the guiding principle of this thesis. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. In the implementation, great emphasis is put on speed, i.e., overall execution speed and convergence speed, while not sacrificing on the freedom to adapt many important numerical details at runtime and not at the compilation stage. This means that the user can test and choose wavelets perfectly suitable for any specific task without having to rebuild the software. The computational overhead of these freedoms is minimized by caching any interim data, e.g., values for the preconditioners and polynomial representations of wavelets in multiple dimensions. Exploiting the structure in the construction of wavelet spaces prevents this step from becoming a burden on the memory requirements while at the same time providing a huge performance boost because necessary computations are only executed as needed and then only once. The essential BVP boundary conditions are enforced using trace operators, which leads to a saddle point problem formulation. This particular treatment of boundary conditions is very flexible, which especially useful if changing boundary conditions have to be accommodated, e.g., when iteratively solving control problems with Dirichlet boundary control based upon the herein considered PDE operators. Another particular feature is that saddle point problems allow for a variety of different geometrical setups, including fictitious domain approaches. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. Local transformations of the wavelet basis are employed to lower the absolute condition number of the already optimally preconditioned operators. The effect of these basis transformations can be seen in the absolute runtimes of solution processes, where the semilinear PDEs are solved as fast as in fractions of a second. This task can be accomplished using simple Richardson-style solvers, e.g., the method of steepest descent, or more involved solvers like the Newton's method. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of semilinear PDE sub-problems. The efficiency of different numerical methods is compared and the theoretical optimal convergence rates and complexity estimates are verified. In summary, this thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve semilinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory

Wavelet based solution of flow and diffusion problems in digital materials

The computation of physical properties in a digital materials laboratory requires significant computational resources. Due to the complex nature of the media, one of the most difficult problems to solve is the multiphase flow problem, and traditional methods such as Lattice Boltzmann are not attractive as the computational demand for the solution is too high. A wavelet based algorithm reduces the amount of information required for computation. Here we solve the Poisson equation for a large three dimensional data set with a second order finite difference approximation. Constraints and fictitious domains are used to capture the complex geometry. We solve the discrete system using a discrete wavelet transform and thresholding. We show that this method is substantially faster than the original approach and has the same order of accuracy. References Mark A. Knackstedt, Shane Latham, Mahyar Madadi, Adrian Sheppard, Trond Varslot, and Christoph Arns, Digital rock physics: 3D imaging of core material and correlations to acoustic and flow properties, The Leading Edge 28 (2009), no. 1, 28--33. Angela Kunoth, Wavelet techniques for the fictitious domain Lagrange multiplier approach, Numer Algorithms 27 (2001), 291--316, doi:10.1023/A:1011891106124 Arthur Sakellariou, Christoph H. Arns, Adrian P. Sheppard, Rob M. Sok, Holger Averdunk, Ajay Limaye, Anthony C. Jones, Tim J. Senden, and Mark A. Knackstedt, Developing a virtual materials laboratory, Mater Today 10 (2007), no. 12, 44--51. Adrian P. Sheppard, Rob M. Sok, and Holger Averdunk, Techniques for image enhancement and segmentation of tomographic images of porous materials, Physica A 339 (2004), no. 1--2, 145--151

Wavelet based solution of flow and diffusion problems in digital materials

The computation of physical properties in a digital materials laboratory requires significant computational resources. Due to the complex nature of the media, one of the most difficult problems to solve is the multiphase flow problem, and traditional methods such as Lattice Boltzmann are not attractive as the computational demand for the solution is too high. A wavelet based algorithm reduces the amount of information required for computation. Here we solve the Poisson equation for a large three dimensional data set with a second order finite difference approximation. Constraints and fictitious domains are used to capture the complex geometry. We solve the discrete system using a discrete wavelet transform and thresholding. We show that this method is substantially faster than the original approach and has the same order of accuracy. References Mark A. Knackstedt, Shane Latham, Mahyar Madadi, Adrian Sheppard, Trond Varslot, and Christoph Arns, Digital rock physics: 3D imaging of core material and correlations to acoustic and flow properties, The Leading Edge 28 (2009), no. 1, 28--33. Angela Kunoth, Wavelet techniques for the fictitious domain Lagrange multiplier approach, Numer Algorithms 27 (2001), 291--316, doi:10.1023/A:1011891106124 Arthur Sakellariou, Christoph H. Arns, Adrian P. Sheppard, Rob M. Sok, Holger Averdunk, Ajay Limaye, Anthony C. Jones, Tim J. Senden, and Mark A. Knackstedt, Developing a virtual materials laboratory, Mater Today 10 (2007), no. 12, 44--51. Adrian P. Sheppard, Rob M. Sok, and Holger Averdunk, Techniques for image enhancement and segmentation of tomographic images of porous materials, Physica A 339 (2004), no. 1--2, 145--151

Adaptive Wavelet Representation And Differentiation On Block-structured Grids

This paper considers a new adaptive wavelet solver for two-dimensional systems based on an adaptive block refinement (ABR) method that takes advantage of the quadtree structure of dyadic blocks in rectangular regions of the plane. The computational domain is formed by non-overlapping blocks. Each block is a uniform grid, but the step size may change from one block to another. The blocks are not predetermined, but they are dynamically constructed according to the refinement needs of the numerical solution. The decision over whether a block should be refined or unrefined is taken by looking at the magnitude of wavelet coefficients of the numerical solution on such block. The wavelet coefficients are defined as differences between values interpolated from a coarser level and known function values at the finer level. The main objective of this paper is to establish a general framework for the construction and operation on such adaptive block-grids in 2D. The algorithms and data structure are formulated by using abstract concepts borrowed from quaternary trees. This procedure helps in the understanding of the method and simplifies its computational implementation. The ability of the method is demonstrated by solving some typical test problems. © 2003 IMACS. Published by Elsevier B.V. All rights reserved.4703/04/15421437Cohen, A., Wavelet methods in numerical analysis (2000) Handbook of Numerical Analysis, 7. , P.G. Ciarlet, LionsJ.L. Amsterdam: ElsevierHolmström, M., (1997) Wavelet Based Methods for Time Dependent PDEs, , Ph.D. Thesis, Uppsala UniversityWalden, J., A general adaptive solver for hyperbolic PDEs based on filter bank subdivisions (2000) Appl. Numer. Math., 33 (1-4), pp. 317-325Vasilyev, O.V., Browman, C., Second generation wavelet collocation method for the solution of partial differential equations (2000) J. Comput. Phys., 165, pp. 660-693Knuth, D.E., (1997) The Art of Programming, , Reading, MA: Addison-WesleyHunter, G.M., Steiglitz, K., Operations on images using quad trees (1979) IEEE Trans. Pattern Anal. Mach. Intell., PAMI-1 (2), pp. 145-153Tromper, R.A., Verwer, J.G., Runge-Kutta methods and local uniform grid refinement (1993) Math. Comput., 60 (202), pp. 591-616Bacry, E., Mallat, S., Papanicolau, G., A wavelet based space-time adaptive numerical method for partial equations (1992) Math. Model. Numer. Anal., 26 (7), pp. 793-834Lötstedt, P., Söderberg, S., Ramage, A., Hemmingsson-Frändén, L., Implicit solution of hyperbolic equations with space-time adaptivity (2002) BIT, 42, pp. 134-158Glowinski, R., Pan, T.-W., Périaux, J., A fictitious domain method for Dirichlet problem and application (1994) Comput. Methods Appl. Mech. Engrg., 111, pp. 283-303Koshigoe, H., Kitahara, K., Finite difference method with fictitious domain applied to a Dirichlet problem (2001) 12th Conference on Domain Decomposition Methods, pp. 151-163. , T. Chan, T. Kako, H. Kawarada, & O. Pironneau. DDM.orgKunoth, A., Wavelet techniques for the fictitious domain - Lagrange multiplier approach (2001) Numer. Algorithms, 27, pp. 291-316Rieder, A., Embedding and a priori wavelet-adaptivity for Dirichlet problems (2000) Modél. Math. Anal. Numér., 34 (6), pp. 1189-120