General Elliptic Boundary Value Problems Research Articles

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Bifurcation of solutions of $U(1)$-equivariant semilinear boundary value problems

Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.

Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity

Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments

The solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. The corner singularity theory for the solutions of elliptic boundary value problems on domains with corners or edges has been well established in the past century and also in recent years. The corner singularity functions provide an appropriate mathematical structure to understand the physical trajectories of the fluid particles. It has been investigated for general elliptic boundary value problems and also extended to some non-elliptic problems. Currently, the theory has been constructed for compressible viscous Stokes and NavierStokes systems on polygonal and polyhedral domains to analyze the structure of the solution near the corners and edges. Several interesting results about the regularity of the solution cannot be extended if one of the following situations appears: The domain has corners, edges and cusp, etc. On the boundary, change of boundary conditions at some points, discontinuities of the solutions, and singularities of the coefficients. This article reviewed the structure of the solution and regularity results of the stationary Stokes and Navier-Stokes equations on polygonal domains with convex or non-convex corners.

Hörmander spaces on manifolds, and their application to elliptic boundaryvalue problems

We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by inner-product H\"ormander spaces for which an RO-varying radial function serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of inner-product Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm property of a general elliptic boundary-value problem on appropriate H\"ormander spaces and find sufficient conditions under which its generalized solutions belong to the space of $p\geq0$ times continuously differential functions.

An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples.

Segmental Refinement: A Multigrid Technique for Data Locality

We investigate a domain decomposed multigrid technique, segmental refinement, for solving general nonlinear elliptic boundary value problems. Brandt and Diskin first proposed this method in 1994; we continue this work by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid can be eliminated on fine grids with modest amounts of extra work and storage while maintaining the asymptotic exactness of full multigrid, although we observe a dependence on an additional parameter not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores of a Cray XC30.

Загальна еліптична крайова задача в розширеній цоболєвській шкалі

Загальна еліптична крайова задача в розширеній цоболєвській шкалі

Superconvergence patch recovery for the gradient of the tensor-product linear triangular prism element

In this article, we study superconvergence of the finite element approximation to the solution of a general second-order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise tensor-product linear triangular prism elements. First, we give the superclose property of the gradient between the finite element solution and the interpolant Πu. Second, we introduce a superconvergence recovery scheme for the gradient of the finite element solution. Finally, superconvergence of the recovered gradient is derived.

On a method for computing waveguide scattering matrices in the presence of point spectrum

A waveguide occupies a domain G in ℝn+1, n ⩾ 1, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter µ. As an approximation to a row of the scattering matrix S(µ) we suggest a minimizer of a quadratic functional JR(·, µ). To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance R, the waveguide outlets to infinity. It is proved that, if a finite interval [µ1, µ2] of the continuous spectrum contains no thresholds, then, as R → ∞, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to µ ∈ [µ1, µ2]. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity.

Radiation and scattering in domains with periodic waveguides under slow stabilization of characteristics of a medium

General elliptic boundary value problems with spectral parameter are studied in a domain $ G \subset {\mathbb{R}^{n + 1}} $ with several “quasicylindrical exits” at infinity. The problems are selfadjoint relative to the Green formula. The coefficients of the boundary value problems are smooth and are slowly stabilized at each exit at infinity to functions that are periodic with respect to the axis t-coordinate. We describe the asymptotics of solutions at infinity. We formulate a well-posed problem with radiation conditions, introduce a unitary scattering matrix, and justify a method for computing this matrix. The results are new even in the particular case where the role of “quasicylinders” are played by usual cylinders (with a constant cross-section) and the coefficients are stabilized at infinity to functions independent of the axis coordinate. Bibliography: 12 titles.

Computational aspects of the local discontinuous Galerkin method on unstructured grids in three dimensions

Using tensor notation, a simplified description of the most relevant operators of the local discontinuous Galerkin (LDG) method applied to a general elliptic boundary value problem on unstructured meshes in three dimensions is presented. A reduction of storage is achieved by introducing a fast algorithm for the assembly of the Schur complement. A semi-algebraic multilevel preconditioner for low-order approximations using the classical Lagrange interpolatory basis is discussed. A series of numerical experiments is presented to illustrate the performance of the proposed preconditioning technique and accuracy of the method on three-dimensional problems.

High accuracy analysis of tensor-product linear pentahedral finite elements for variable coefficient elliptic equations

For a general second-order variable coefficient elliptic boundary value problem in three dimensions, the authors derive the weak estimate of the first type for tensor-product linear pentahedral finite elements. In addition, the estimate for the W 1, 1-seminorm of the discrete derivative Green’s function is given. Finally, the authors show that the derivatives of the finite element solution u h and the corresponding interpolant Πu are superclose in the pointwise sense of the L ∞-norm.

Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions

Fixed domain methods have well-known advantages in the solution of variable domain problems including inverse interface problems. This paper examines two new control approaches to optimal design problems governed by general elliptic boundary value problems with Dirichlet boundary conditions. Numerical experiments are also included.

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.

Generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions

We introduce a method for approximating essential boundary conditions—conditions of Dirichlet type—within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form - ∑ i , j = 1 n ( a ij u x i ) x j + ∑ i = 1 n b i u x i + cu = f in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, { S μ } μ ⩾ 1 , which are nonconforming (that is S μ ⊄ H 0 1 ( Ω ) ). We assume that ∥ v ∥ L 2 ( ∂ Ω ) ⩽ Ch μ m ∥ v ∥ H 1 ( Ω ) , for all v ∈ S μ , and there exists u I ∈ S μ such that ∥ u - u I ∥ H 1 ( Ω ) ⩽ Ch μ j ∥ u ∥ H j + 1 ( Ω ) , 0 ⩽ j ⩽ m , where u ∈ H m + 1 ( Ω ) is the exact solution, m is the expected order of approximation, and h μ is the typical size of the elements defining S μ . Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence u μ ∈ S μ of u. Next, we extend our analysis to the inhomogeneous boundary value problem - ∑ i , j = 1 n ( a ij u x i ) x j + ∑ i = 1 n b i u x i + cu = f in Ω , u = g on ∂ Ω . Finally, we outline the construction of a sequence of GFEM spaces S μ ⊂ S ˜ μ , μ = 1 , 2 , … , that satisfies our assumptions.

Locally-corrected spectral methods and overdetermined elliptic systems

We present fast locally-corrected spectral methods for linear constant-coefficient elliptic systems of partial differential equations in d-dimensional periodic geometry. First, arbitrary second-order elliptic systems are converted to overdetermined first-order systems. Overdetermination preserves ellipticity, while first-order systems eliminate mixed derivatives, resolve convection–diffusion conflicts, and simplify derivative computations. Second, a periodic fundamental solution is derived by Fourier analysis and mollified for rapid convergence, independent of the regularity of the elliptic problem. Third, a new Ewald summation technique for first-order elliptic systems locally corrects the mollified solution to achieve high-order accuracy. We also discuss second-kind boundary integral equations based on single layer potentials formed with the mollified and corrected fundamental solution, which form a useful toolkit for solving general elliptic boundary value problems in general domains. The resulting spectral methods provide highly accurate solutions and derivatives for periodic problems.

On the Root Functions of General Elliptic Boundary Value Problems

We consider a boundary value problem for an elliptic differential operator of order 2m in a domain \({\mathcal{D}} \subset {\mathbb{R}}^{n}\). The boundary of \({\mathcal{D}}\) is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and \(\partial {\mathcal{D}}\) bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of \(\partial {\mathcal{D}}\). The corresponding spaces are weighted Sobolev spaces \(H^{{s,\gamma }} {\left( {\mathcal{D}} \right)}\), and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in \(L^{2} ({\mathcal{D}})\). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.

Analyticity of solutions of analytic non-linear general elliptic boundary value problems, and some results about linear problems

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L p type are obtained.

Self–adjoint Extensions for the Neumann Laplacian and Applications

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so–called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self–adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains. The error estimates for proposed approximations of shape functionals are provided.