Order Elliptic Boundary Value Problems Research Articles

Nonzero positive solutions of systems of elliptic boundary value problems

A new result on existence of nonzero positive solutions of systems of second order elliptic boundary value problems is obtained under some sublinear conditions involving the principle eigenvalues of the corresponding linear systems. Results on eigenvalue problems of such elliptic systems are derived and generalize some previous results on the eigenvalue problems of systems of Laplacian elliptic equations. Applications of our results are given to two such systems with specific nonlinearities.

EXISTENCE OF NONZERO POSITIVE SOLUTIONS OF SYSTEMS OF SECOND ORDER ELLIPTIC BOUNDARY VALUE PROBLEMS

Existence of nonzero positive solutions of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a flxed point equation in real Banach spaces by using the wellknown theory of flxed point index for compact maps deflned on cones, where the flxed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in ordered Banach spaces.

Uniqueness in some higher order elliptic boundary value problems in n dimensional domains

We develop maximum principles for several P functions which are defined on solutions to equations of fourth and sixth order (including a equation which arises in plate theory and bending of cylindrical shells). As a consequence, we obtain uniqueness results for fourth and sixth order boundary value problems in arbitrary n dimensional domains.

A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence

We investigate a conjecture regarding the number of solutions of asecond order elliptic boundary value problem with an asymmetricnonlinearity. This investigation makes use of several computerassisted techniques. First, we compute approximate solutions usingNewton's Iteration for small $b$ and then use a continuation methodto show that the number of solutions becomes large as $b$ increases.

Two-Level Additive Schwarz Preconditioners for a Weakly Over-Penalized Symmetric Interior Penalty Method

We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.

Elliptic boundary value problems with λ-dependent boundary conditions

In this paper second order elliptic boundary value problems on bounded domains Ω ⊂ R n with boundary conditions on ∂ Ω depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz–Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space L 2 ( Ω ) ⊕ ( L 2 ( ∂ Ω ) ) m , which is a natural generalization of known results on λ-linear elliptic boundary value problems and λ-rational boundary value problems for ordinary second order differential equations.

Optimality of local multilevel methods on adaptively refined meshes for elliptic boundary value problems

A local multilevel product algorithm and its additive version are analyzed for linear systems arising from the application of adaptive finite element methods to second order elliptic boundary value problems. The abstract Schwarz theory is applied to verify uniform convergence of local multilevel methods featuring Jacobi and Gauß–Seidel smoothing only on local nodes. By this abstract theory, convergence estimates can be further derived for the hierarchical basis multigrid method and the hierarchical basis preconditioning method on locally refined meshes, where local smoothing is performed only on new nodes. Numerical experiments confirm the optimality of the suggested algorithms.

The optimal convergence rate of a [formula omitted] finite element method for non-smooth domains

We establish optimal (up to arbitrary ε > 0 ) convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weighted H 2 Sobolev space with 5th degree Argyris elements. This formulation arises while generalizing to the case of non-smooth domains an unconditionally stable scheme developed by Liu et al. [J.-G. Liu, J. Liu, R.L. Pego, Stability and convergence of efficient Navier–Stokes solvers via a commutator estimate, Comm. Pure Appl. Math. 60 (2007) pp. 1443–1487] for the Navier–Stokes equations. We prove the optimality for both quasiuniform and graded mesh refinements, and provide numerical results that agree with our theoretical predictions.

Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems

We consider the 2 m-th order elliptic boundary value problem L u = f ( x , u ) on a bounded smooth domain Ω ⊂ R N with Dirichlet boundary conditions on ∂ Ω. The operator L is a uniformly elliptic linear operator of order 2 m whose principle part is of the form ( − ∑ i , j = 1 N a i j ( x ) ∂ 2 ∂ x i ∂ x j ) m . We assume that f is superlinear at the origin and satisfies lim s → ∞ f ( x , s ) s q = h ( x ) , lim s → − ∞ f ( x , s ) | s | q = k ( x ) , where h , k ∈ C ( Ω ¯ ) are positive functions and q > 1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.

Locally optimized [formula omitted] preconditioning of Rannacher–Turek FEM systems

In this paper Rannacher–Turek non-conforming rotated bilinear finite elements are applied for the numerical solution of second order elliptic boundary value problems. The preconditioned conjugate gradient method is used for the iterative solution of the arising linear algebraic system A u = f . A locally optimized construction for an M-matrix approximation B of the global stiffness matrix A is the first step of the proposed algorithm. Then, the preconditioner is obtained by modified incomplete Cholesky factorization of the auxiliary M-matrix B. A comparative analysis concerning three different approaches for construction of such matrices B is presented. The related spectral condition number estimates are derived. The most important contributions of the paper is the developed original robust preconditioning scheme for strongly anisotropic problems based on properly skewed meshes. A set of numerical tests is presented to illustrate the theoretical investigations.

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains \({\Omega \subset\mathbb{R}^n}\) , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided \({n\geqq 3}\) . Moreover, the biharmonic Green’s function in balls \({B\subset\mathbb{R}^n}\) under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for \({n\geqq 3}\) .

The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Assume that the elliptic operator L=div (A(x)∇) is Lp-resolutive, p>1, on the unit disc \(\mathbb{D}\subset \mathbb {R}^{2}\) . This means that the Dirichlet problem $$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\end{array}\right.$$ is uniquely solvable for any \(g\in L^{p}(\partial\mathbb{D})\) . Then, there exists e>0 such that L is Lr- resolutive in the optimal range p−e<r≤∞ (Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Conference Board of the Mathematical Sciences, vol. 83, Am. Math. Soc., Providence, 1991). Here we determine the precise value of e in terms of p and of a natural “norm” of the harmonic measure ωL.

The BPS preconditioner on Beowulf cluster

This work presents the implementation on a Linux Cluster of a parallel preconditioner for the solution of the linear system resulting from the finite element discretization of a 2D second order elliptic boundary value problem. The numerical method, proposed by Bramble, Pasciak and Schatz, is developed using Domain Decomposition techniques, which are based on the splitting of the computational domain into subregionsof smaller size, enforcing suitable compatibility conditions. The Fortran code is implemented using PETSc: a suite of data structures and routines devoted to the scientific parallel computing and based on the MPI standard for all message-passing communications. The main interest of the paper is to present an efficient and portable code for the solution of large-scale linear systems and to investigate how the architecturalaspects of the cluster influence the performance of the considered algorithm. We provide an analysis of the execution times as well as of the scalability, using as test case the classical Poisson equation with Dirichlet boundary conditions.

On the solvability of the higher order elliptic boundary value problem

In this paper, by employing a non-variational version of a max–min principle, new results on existence and uniqueness of generalized solution to the boundary value problem of higher order elliptic partial differential equations at resonance are given.

A new approach to Richardson extrapolation in the finite element method for second order elliptic problems

This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in R N \mathbb R^N , N ≥ 2 N \ge 2 . The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given.

Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n , the negative part of the corresponding Green's function is “small” when compared with its singular positive part, provided that n ⩾ 3 . To cite this article: H.-Ch. Grunau, F. Robert, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems

We provide an a posteriori error analysis of finite element approximations of pointwise state constrained distributed optimal control problems for second order elliptic boundary value problems. In particular, we derive a residual-type a posteriori error estimator and prove its efficiency and reliability up to oscillations in the data of the problem and a consistency error term. In contrast to the case of pointwise control constraints, the analysis is more complicated, since the multipliers associated with the state constraints live in measure spaces. The analysis essentially makes use of appropriate regularizations of the multipliers both in the continuous and in the discrete regime. Numerical examples are given to illustrate the performance of the error estimator.

Existence and uniqueness of solutions for higher order elliptic boundary value problems

A class of sufficient conditions are obtained for the existence and uniqueness of solutions to the boundary value problems of semi-linear elliptic partial differential equations, using a global inverse function theorem.

Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method

We study the convergence of an adaptive interior penalty discontinuous Galerkin (IPDG) method for a two-dimensional model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh-dependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods [O. Karakashian and F. Pascal, Convergence of Adaptive Discontinuous Galerkin Approximations of Second-order Elliptic Problems, preprint, University of Tennessee, Knoxville, TN, 2007], the convergence analysis does not require multiple interior nodes for refined elements of the triangulation. In fact, it will be shown that bisection of the elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. The results of numerical experiments are given to illustrate the performance of the adaptive method.