We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes.We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as (Tz⊗I + I⊗Axy) U=F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems. 相似文献
The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form in a real Hilbert space ℋ︁ with a semi‐bounded self‐adjoint operator A0, while for every y from a dense subspace X of ℋ︁, B(y ) is a symmetric operator. The left‐hand side is assumed to be related to a certain auxiliary functional ψ, and the associated linear problems are supposed to have non‐empty discrete spectrum (y ∈ X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (∗︁) on a sphere SR ≔ {y ∈ X | ∥y∥ℋ︁ = R} whose ψ‐value is the n‐th Ljusternik‐Schnirelman level of ψ| and whose corresponding eigenvalue is the n‐th eigenvalue of the associated linear problem (∗︁∗︁), where R > 0 and n ∈ ℕ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an n‐th eigenfunction of a linear problem of the form (∗︁∗︁). We discuss applications to elliptic partial differential equations with radial symmetry. 相似文献
This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R 3. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points . 相似文献
We consider a discretization of linear elliptic boundary value problems in 2-D by the new version of the mortar finite element method which uses locally nonconforming Crouzeix-Raviart elements. We show that if a solution of the original differential problem belongs to the space H2(), then an error is of the same order as in the standard nonconforming finite element method. We also propose an additive Schwarz method of solving the discrete problem and show that its rate of convergence is almost optimal. 相似文献
The electric potential u in a solution of an electrolyte around a linear polyelectrolyte of the form of a cylinder satisfies We study the problem when R → ∞. 相似文献
Let D ? ?n be a bounded domain with piecewise-smooth boundary, and q(x,t) a smooth function on D × [0, T]. Consider the time-like Cauchy problem Given g, h for which the equation has a solution, we show how to approximate u(x,t) by solving a well posed fourth-order elliptic partial differential equation (PDE). We use the method of quasi-reversibility to construct the approximating PDE. We derive error estimates and present numerical results. 相似文献
We consider the numerical solution of a c‐stable linear equation in the tensor product space , arising from a discretized elliptic partial differential equation in . Utilizing the stability, we produce an equivalent d‐stable generalized Stein‐like equation, which can be solved iteratively. For large‐scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of computational complexity, under appropriate assumptions (with ns being the flop count for solving a linear system associated with ). Illustrative numerical examples will be presented. 相似文献
In this article we study a controllability problem for an elliptic partial differential equation in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution, with a given right hand side source term, into an open subdomain. The mapping that associates this trace to the shape of the domain is nonlinear. We first consider the linearized problem and show an approximate controllability property. We then address the same questions in the context of a finite difference discretization of the elliptic problem. We prove a local controllability result applying the Inverse Function Theorem together with a ``unique continuation' property of the underlying adjoint discrete system.
Mathematics Subject Classification (1991):35J05, 93B03, 65M06 相似文献
We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations. 相似文献
In this paper we give a necessary and sufficient condition for the oscillation of the second order linear differential equation where p is a locally integrable function and either or where We give some applications which show how these results unify and imply some classical results in oscillation theory. 相似文献
In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R3 with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S1 of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space Wpl(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2). 相似文献
Summary. We formulate elliptic boundary value problems with stochastic loading in a bounded domain Dd. We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation.
Mathematics Subject Classification (2000): 65N30Research performed under IHP network Breaking Complexity of the EC, contract number HPRN-CT-2002-00286, and supported in part by the Swiss Federal Office for Science and Education under grant number BBW 02.0418. 相似文献
We prove using the Faedo-Galerkin method the existence of a generalized solution of an initial-boundary value problem for the non-linear evolution equation 0 ? Q ? 2, in a cylinder QT = Ω × (0, T), where ?? u = yuxx + uyy is the Tricomi operator and l(u) a special differential operator of first order. We then show that the approximate generalized solution of problem (*) converges to the approximate generalized solution of the corresponding stationary boundary value problem as t → ∞. 相似文献