Wavelet Techniques for the Fictitious-Domain-Lagrange-Multiplier-Approach

This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments of wavelet methods, and a natural way of preconditioning the resulting systems of linear equations. We describe first a general format of variational problems that are well-posed in a certain natural topology. In order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems, their formulation as a first order system, transmission problems, the system of Stokes equations, or more general saddle point problems. Particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context. This provides access to "difficult" norms, efficient preconditioners and, in the case of first order systems, optimal L2 error estimates.

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

Scaling function and wavelet preconditioners for second order elliptic problems

(1981). Second order elliptic boundary value problems in a domain with edges. Communications in Partial Differential Equations: Vol. 6, No. 9, pp. 1023-1042.

The Variational Problems and Equivalent Forms for a Second Order Elliptic Boundary Value Problem

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only e2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the e2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization. In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level. Finally, numerical results are provided.

We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.

Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting interface control domain decomposition method is robust with respect to coefficients variations; it can exploit nonconforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advection--advection-diffusion couplings.

Second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered. The behavior of the ordinary Galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed.

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N \mathbb {R}^{N} . In a sense to be discussed below these sharpen known quasi–optimal L ∞ L_{\infty } and W ∞ 1 W^{1}_{\infty } estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u u . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in R N \mathbb {R}^{N} and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems.

Boundedness and continuity of solutions to second order elliptic boundary value problems

We derive an optimal multilevel preconditioner for nonconforming P1 element discretizations of second order elliptic boundary value problems.

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

SummaryThe main result in this paper is that the solution operator for the bi-Laplace problem with zero Dirichlet boundary conditions on a bounded smooth 2d-domain can be split in a positive part and a possibly negative part which both satisfy the zero boundary condition. Moreover, the positive part contains the singularity and the negative part inherits the full regularity of the boundary. Such a splitting allows one to find a priori estimates for fourth order problems similar to the one proved via the maximum principle in second order elliptic boundary value problems. The proof depends on a careful approximative fill-up of the domain by a finite collection of limaçons. The limaçons involved are such that the Green function for the Dirichlet bi-Laplacian on each of these domains is strictly positive.

The original additive average Schwarz method and its variants were introduced for solving second order elliptic boundary value problems. In this paper, we extend the original idea to fourth order problems by designing a variant of the additive average Schwarz method for the plate bending problem using the nonconforming Morley finite element for the discretization. Like the original average method this method uses only nonoverlapping subdomains, and its coarse space does not need any explicit coarse mesh. An analysis of the method, and some numerical experiments to validate the theory are presented.