Second-order Elliptic Boundary Value Problems Research Articles

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I

First-order system least squares (FOSLS) is a recently developed methodology for solving partial differential equations. Among its advantages are that the finite element spaces are not restricted by the inf-sup condition imposed, for example, on mixed methods and that the least-squares functional itself serves as an appropriate error measure. This paper studies the FOSLS approach for scalar second-order elliptic boundary value problems with discontinuous coefficients, irregular boundaries, and mixed boundary conditions. A least-squares functional is defined, and ellipticity is established in a natural norm of an appropriately scaled least-squares bilinear form. For some geometries, this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. A companion paper shows that the singular basis functions can be added at little extra cost and lead to optimal performance of standard finite element discretization and multilevel solver techniques, also independent of the size of coefficient jumps for some geometries.

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part II

First-order system least squares (FOSLS) is a methodology that offers an alternative to standard methods for solving partial differential equations. This paper studies the first-order system least-squares approach for scalar second-order elliptic boundary value problems with discontinuous coefficients. In a companion paper [M. Berndt, T. A. Manteuffel, S. F. McCormick, and G. Starke, Analysis of first-order system least squares ({FOSLS}) for elliptic problems with discontinuous coefficients: Part I, SIAM J. Numer. Anal., 43 (2005), pp. 386--408], ellipticity of an appropriately scaled least-squares bilinear form is established in a natural norm. For some geometries this ellipticity is independent of the size of the jumps in the coefficients. The occurrence of singularities at interface corners, cross-points, reentrant corners, and irregular boundary points is discussed, and a basis of singular functions with local support around singular points is established. This paper describes a method for including discrete versions of the singular basis functions together with standard finite element spaces in a least-squares format at little additional computational cost. The singular basis functions are constructed to match the jump conditions that arise at interfaces between regions of continuity of the diffusion coefficient. Because these basis functions must be approximated in practice, the resulting discretization is by nature nonconforming. This necessitates the establishment here of a general error estimate for FOSLS L2 minimization problems discretized by nonconforming finite elements. An advantage of the FOSLS formulation is that this estimate does not involve the consistency error term usually present in bounds for other methods. Based on this general estimate, error bounds are derived for the finite element space that includes singular basis functions. Numerical tests are included that confirm these discretization error bounds. Finally, a multilevel method is developed for solving the discrete system that uses singular basis functions on all levels, and its efficiency is demonstrated by the numerical results.

A Multiscale Finite Element Method for Numerical Homogenization

This paper is concerned with a multiscale finite element method for numerically solving second-order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and a fine local mesh, the latter being used for computing independently an adapted finite element basis for the coarse mesh. The main idea is the introduction of a composition rule, or change of variables, for the construction of this finite element basis. In particular, this allows for a simple treatment of high-order finite element methods. We provide optimal error estimates in the case of periodically oscillating coefficients. We illustrate our method in various examples.

An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations

The authors and their colleagues have developed numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem using the Newton-like operator with appropriate approximation and constructive a priori error estimates for Poisson's equations. Many verification results show that the authors' methods are sufficiently useful when the equation has no first-order derivative. However, in the case that the equation includes the term of a first-order derivative, there is a possibility that the verification algorithm does not work even though we adopt a sufficiently accurate approximation subspace. The purpose of this paper is to propose an alternative method to overcome this difficulty. Numerical examples which confirm the effectiveness of the new method are presented.

Existence of three solutions for some elliptic equations with jumping nonlinearities

We consider the existence of solutions of a second-order semilinear elliptic boundary value problem with Dirichlet boundary condition, - Δ u = b 1 [ ( u + 1 ) + - 1 ] + b 2 u + in Ω , where Ω is a bounded open set in R N with smooth boundary ∂ Ω . We investigate them in four regions of ( b 1 , b 2 ).

Factorization of second-order elliptic boundary value problems by dynamic programming

We present a method to factorize a second-order boundary value problem in a cylindrical domain in a system of uncoupled first-order initial value problems, together with a nonlinear Riccati-type equation for functional operators. This uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder. This method can be viewed as an infinite-dimensional generalization of the block Gauss LU factorization.

Factorization of second-order elliptic boundary value problems by dynamic programming

We present a method to factorize a second-order boundary value problem in a cylindrical domain in a system of uncoupled first-order initial value problems, together with a nonlinear Riccati-type equation for functional operators. This uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder. This method can be viewed as an infinite-dimensional generalization of the block Gauss LU factorization.

Multigrid preconditioners for symmetric Sinc systems

The symmetric Sinc-Galerkin method applied to a separable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations ( ? x ?D y + D x ?? y ) u = g where ? is the Kronecker product symbol, ? x and ? y are Toeplitz-plus-diagonal matrices, and D x and D y are diagonal matrices. The main contribution of this paper is to present a two-step preconditioning strategy based on the banded matrix approximation and the multigrid iteration for these Sinc-Galerkin systems. Numerical examples show that the multigrid preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system.

A Quasi-Dual Function Method for Extracting Edge Stress Intensity Functions

We present a method for the computation of the coefficients of singularities along the edges of a polyhedron for second-order elliptic boundary value problems. The class of problems considered includes problems of stress concentration along edges or crack fronts in general linear three-dimensional elasticity. Our method uses an incomplete construction of three-dimensional dual singular functions, based on explicitly known dual singular functions of two-dimensional problems tensorized by test functions along the edge and combined with complementary terms improving their orthogonality properties with respect to the edge singularities. Our method is aimed at the numerical computation of the stress intensity functions. It is suitable for a postprocessing procedure in the finite element approximation of the solution of the boundary value problem.

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.

A hybrid preconditioner of banded matrix approximation and alternating direction implicit iteration for symmetric Sinc–Galerkin linear systems

The symmetric Sinc–Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations (Ψ x⊗D y+D x⊗Ψ y) u = g , where⊗ is the Kronecker product symbol, Ψ x and Ψ y are Toeplitz-plus-diagonal matrices, and D x and D y are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc–Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc–Galerkin linear system.

Domain Decomposition for a Mixed Finite Element Method in Three Dimensions

We consider the solution of the discrete linear system resulting from a mixed finite element discretization applied to a second-order elliptic boundary value problem in three dimensions. Based on a decomposition of the velocity space, these equations can be reduced to a discrete elliptic problem by eliminating the pressure through the use of substructures of the domain. The practicality of the reduction relies on a local basis, presented here, for the divergence-free subspace of the velocity space. We consider additive and multiplicative domain decomposition methods for solving the reduced elliptic problem, and their uniform convergence is established.

Existence of solutions for nonlinear elliptic boundary value problems

Hilbert space method is applied to a class of semilinear second-order elliptic boundary value problems and the existence of solutions is obtained with some restrictions.

A nonlinear boundary value problem containing nonstandard boundary conditions

In this paper, we consider a second-order nonlinear elliptic boundary value problem (BVP) on a bounded domain Ω⊂ R N , with nonstandard boundary conditions (BCs). More precisely, at some boundary part, we impose a Dirichlet BC containing an unknown additive constant, accompanied of a nonlocal (integral) Neumann side condition. As an auxiliary problem, we first study the linear case, which, moreover, has practical relevance itself. Leaning upon the results obtained, we prove the existence and uniqueness of a solution to the nonlinear BVP considered, using, in particular, the maximum principle.

A parallel domain decomposition algorithm of mixed element equation for second-order elliptic Dirichlet boundary value problem

A parallel domain decomposition algorithm of Schwarz type is introduced to solve the mixed finite element equation for the Dirichlet boundary value problem of a second-order partial differential equation of elliptic type and convergence of this algorithm is analyzed.

Decoupling Three-Dimensional Mixed Problems Using Divergence-Free Finite Elements

In this paper we describe an iterative method for indefinite saddle-point systems arising from mixed finite element discretizations of second-order elliptic boundary value problems subject to mixed boundary conditions and posed over polyhedral three-dimensional domains. The method is based on a decoupling of the vector of velocities in the saddle-point system from the vector of pressures, resulting in a symmetric positive definite velocity system and a triangular pressure system. The crucial step in this approach is the construction of the divergence-free Raviart--Thomas--Nedelec elements from the curls of Nedelec's edge elements. Because of the large kernel of the curl-operator, this representation is not unique. To find a basis we consider the graph made up of the nodes and edges of the mesh and eliminate the edge elements associated with a spanning tree in this graph. To prove that this technique works in the general case considered here, we employ fundamental results from algebraic topology and graph theory. We also include some numerical experiments, where we solve the (decoupled) velocity system by ILU-preconditioned conjugate gradients and the pressure system by simple back substitutions. We compare our method with a standard ILU-based block preconditioner for the original saddle-point system, and we find that our method is faster by a factor of at least 4.5 in all cases, with the greatest improvement occurring in the nonuniform mesh case.

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Convergence of the multigrid $V$-cycle algorithm for second-order boundary value problems without full elliptic regularity

The multigrid V-cycle algorithm using the Richardson relaxation scheme as the smoother is studied in this paper. For second-order elliptic boundary value problems, the contraction number of the V-cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps, without assuming full elliptic regularity. As a consequence, the V-cycle convergence result of Braess and Hackbusch is generalized to problems without full elliptic regularity.

NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS

In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques for self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author's own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article.

Discrete boundary conditions for problems with interface

A discrete boundary condition on an artificial boundary is obtained for some second-order elliptic boundary value problems with interface. The problem is reduced to a problem without singularities, and the standard finite element method is used to solve this problem. Some numerical examples are given.