Boundary value problems for elliptic partial differential operators on bounded domains

This paper deals with a new mixed finite element solution of the Dirichlet problem of fourth order elliptic linear partial differential operators with variable coefficients. As examples, bending problems of elastic plates having variable thickness have been considered, for which the algorithm of this method allows a simultaneous determination of the deflection, the curvature components and the „actual”︁ bending and twisting moments of the plate, the „actual”︁ bending and twisting moments being the components of the Lagrange multiplier of this method. Suitable error estimates are obtained under necessary regularity assumptions on the solution.

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.

For the numerical solution of linear variational problems involving elliptic partial differential operators in n ≥ 2 space dimensions, iterative solution schemes are indispensable on account of their problem size. Our guiding principle is to devise iterative solvers which are optimal in the number of arithmetic operations, i.e., which are of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are required. The class of preconditioners for which this can be shown are of multilevel type, requiring nested approximation spaces to approximate the solution of the system on a fine user-specified grid. For smooth solutions of standard second and fourth order elliptic PDEs (partial differential equations) in variational form, approximations based on tensor products of higher-order B-splines yield high accuracy.For such problem classes, this survey collects the main ingredients for multilevel preconditioners in terms of higher order B-splines. There are three types of multilevel preconditioners for which asymptotic optimality can be shown. Two of them, the so-called additive preconditioners, are specified for isogeometric analysis involving linear elliptic partial differential operators in terms of variants of the BPX (Bramble-Pasciak-Xu) preconditioner and wavelet preconditioners. The third class are the so-called multiplicative preconditioners, specifically, multigrid methods.An essential ingredient for all these multilevel preconditioners are intergrid operators which transform vectors or matrices between grids of different grid spacing. For higher order B-splines, these intergrid operators can be derived from their refinement relations. In addition to a presentation of the theoretical ingredients, the performance of the different preconditioners will be demonstrated by some numerical examples.KeywordsVariational InequalityRiesz BasisBiorthogonal WaveletElliptic PDEsSpline SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the study of elliptic differential operators of second order, we found that the automorphism pseudogroups are of finite type ([4]). However this fact takes a complete change in elliptic differential operators of first order.So as to make the objects which can be dealt with clear, we introduce the concept of pseudoellipticity of first order differential operators (Definition 1.1), which is naturally satisfied by first order elliptic differential operators.

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operatorsPof order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equationPu=fin the Sobolev space Wp,2(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of aP‐Neumann problem for elliptic differential operators is proved.

Let {λ2} and {ϕ λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by\(R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f(} \lambda ) \varphi _\lambda \), are uniformly bounded from the Hardy spaceH p (M) into Weak-L p (M), if 0<p<1 and δ=N/p−(N+1)/2.

For a linear second order elliptic partial differential operator A : V → V′, we consider the boundary value problems Au = f with stationary Gaussian random data f over the dual V′ of the separable Hilbert space V in which the solution u is sought. The operator A is assumed to be deterministic and bijective. The unique solution u = A−1f is a Gaussian random field over V. It is characterized by its mean field Eu and its covariance Cu ∈ V ⊗ V. For a class of Gaussian data f with piecewise analytic covariance kernels Cf ∈ V′ ⊗ V′, we characterize analytic regularity of the covariance Cu of the Gaussian solution u in families of countably normed, weighted Sobolev spaces. To this end, we investigate shift theorems for the (nonhypoelliptic) deterministic tensor PDEs (A ⊗ A)Cu = Cf proposed by Schwab and Todor, (Numer Math 95 (2003), 707–734) for the computation of covariance Cu of the random solution u. The nonhypoelliptic nature of A ⊗ A implies that sing supp(Cu) is in general strictly larger than sing supp(Cf). For a model problem, analyticity and singular support of the solution is characterized completely. Based on our regularity results, we outline an hp‐finite element strategy (Pentenrieder, Ph.D thesis, ETHZurich, Dissertation No. 18729, 2009; Pentenrieder and Schwab, Research Report 2010‐08, Seminar for Applied Mathematics, ETH Zürich, Submitted) to approximate Cu stemming from covariances of stationary Gaussian data f. In the second part (Pentenrieder and Schwab, Research Report 2010–08, Seminar for Applied Mathematics, ETH Zürich, Submitted) of this work, we prove that this discretization gives exponential rates of convergence of the FE approximations, in terms of the number of degrees of freedom. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

The geometry which is the topic of this book is that determined by a map of one space N onto another, M, mapping a diffusion process, or operator, on N to one on M. Filtering theory is the science of obtaining or estimating information about a system from partial and possibly flawed of it. The system itself may be random, and the flaws in the can be caused by additional noise. In this volume the randomness and noises will be of Gaussian white noise type so that the system can be modelled by a diffusion process; that is it evolves continuously in time in a Markovian way, the future evolution depending only on the present situation. This is the standard situation of systems governed by Ito type stochastic differential equations. The state space will be the smooth manifold, N, possibly infinite dimensional, and the observations will be obtained by a smooth map onto another manifold, N, say. We emphasise that the geometry is important even when both manifolds are Euclidean spaces. This can also be viewed from a purely partial differential equations viewpoint as one smooth second order elliptic partial differential operator lying above another, both with no zero order term. We consider the geometry of this situation with special emphasis on situations of geometric, stochastic analytic, or filtering interest. The most well studied case is of one Brownian motion being mapped to another with a consequent skew product decomposition (or equivalently the case of Riemannian submersions). This sort of decomposition is generalised and a key to the rest of the book. It is used to study in particular, classical filtering, (semi-)connections determined by stochastic flows, and generalised Weitzenbock formulae.

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.

Let $\\Omega\\subset{\\mathbb R}^n$ be bounded with a smooth boundary $\\Gamma$\nand let $S$ be the symmetric operator in $L^2(\\Omega)$ given by the minimal\nrealization of a second order elliptic differential operator. We give a\ncomplete classification of the Markovian self-adjoint extensions of $S$ by\nproviding an explicit one-to-one correspondence between such extensions and the\nclass of Dirichlet forms in $L^2(\\Gamma)$ which are additively decomposable by\nthe bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form.\nBy such a result two further equivalent classifications are provided: the first\none is expressed in terms of an additive decomposition of the bilinear forms\nassociated to the extensions, the second one uses the additive decomposition of\nthe resolvents provided by Krein's formula. The Markovian part of the\ndecomposition allows to characterize the operator domain of the corresponding\nextension in terms of Wentzell-type boundary conditions. Some properties of the\nextensions, and of the corresponding Dirichlet forms, semigroups and heat\nkernels, like locality, regularity, irreducibility, recurrence, transience,\nultracontractivity and Gaussian bounds are also discussed.\n

When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form ε Σ a i j ∂ 2 / ∂ x i ∂ x i + Σ b i ∂ / ∂ x i \varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i} then results are known regarding the behavior of the principal eigenvalue λ = λ ε \lambda = {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 . These results are very sharp in case the vector ( b i ) ({b_i}) has a unique asymptotically stable point in the domain ω \omega where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of ω \omega is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of λ ε {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 are established.

On preconditioned eigensolvers and Invert–Lanczos processes

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh 0. Every positive L-harmonic function has a unique representation

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.