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Thomas has also very important contributions to the finite element analysis of optimal control problems in particular to error estimates for Finite element discretizations with boundary control, [[14]]. Keywords: Finite Elements; Optimal Control; Error Estimates; Numerical Analysis; 35J05; 35J25; 35J75; 49M05; 49M25; 49M29; 65M10; 65M12; 65M15; 65M60; 65N12; 65N30 EN Finite Elements Optimal Control Error Estimates Numerical Analysis 35J05 35J25 35J75 49M05 49M25 49M29 65M10 65M12 65M15 65M60 65N12 65N30 813 815 3 10/06/23 20231001 NES 231001 Dedicated to Thomas Apel on the occasion of his 60th birthday This special issue of Computational Methods in Applied Mathematics is dedicated to Thomas Apel on the occasion of his 60th birthday in 2022. Thomas worked also together with Olaf Steinbach on Finite element analysis for optimal control problems, [[20]], and Gabriel Acosta on Raviart-Thomas interpolation on anisotropic tetrahedra, [[1]]. Appl. 52 (2012), no. 1, 3-28. 15 T. Apel, J. Pfefferer and M. Winkler, Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra, Math. [Extracted from the article]

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named "adaptive Runge–Kutta regularization", which does not require any penalization term. Finally, we compare them numerically. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss the use of the Linear Sampling Method to reconstruct impenetrable inclusions from Electrostatic Cauchy data. We consider the case of a perfectly conducting and impedance inclusion. In either case, we show that the Dirichlet to Neumann mapping can be used to reconstruct impenetrable sub-regions via a sampling method. We also propose a direct method based on boundary integral equations to reconstruct the impedance parameter using the reconstructed boundary of the inclusion from the knowledge of multiple Cauchy pairs which can be computed from Dirichlet to Neumann mapping. Some numerical reconstructions are presented in two space dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ ⁡ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ (Ω) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In this paper a Banach function space (b.f.s. in short) X (Ω) on a n-dimensional bounded domain Ω with Lebesgue measure is considered. The Banach function space X (S) on the (n − 1) -dimensional surface S ⊂ Ω ¯ generated by the norm of the space X (Ω) is defined. The Sobolev function space W X m (Ω) is defined using the norm of the space X (Ω) , as well as the concept of the trace of an arbitrary function from this space on the surface S. Based on this concept, the space of traces W X m (S) is defined and a characterization of this space is given. It was proved that it is boundedly embedded in the space X (S). Particular cases of a functional space X (Ω) can be Lebesgue spaces, Lebesgue spaces with variable summability exponents, Morrey space, Orlicz space, grand Lebesgue spaces and etc. The obtained results allow us to consider boundary value problems for differential equations in b.f.s. W X m (Ω). [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlocal boundary value problems for the Laplace operator in a ball, which are a multidimensional generalisation of the Samarskii–Ionkin problem. The well-posedness of the problems are investigated, and Fredholm property of the problems are studied. Moreover, we obtain integral representations of their solutions in explicit forms. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a kind of tempered fractional differential equation Riemann–Stieltjes integral boundary value problems with p-Laplacian operators. By means of the sum-type mixed monotone operators fixed point theorem based on the cone P h , we obtain not only the local existence with a unique positive solution, but also construct two successively monotone iterative sequences for approximating the unique positive solution. Finally, we present an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

The question on asymptotic representations of Green's function for classical boundary value problems for second-order elliptic equations depending on a complex parameter is considered. The equations are defined either in a domain with a compact boundary or in the whole space or the half-space. The main asymptotic term is written out. The results can be applied for the study of some inverse problems of recovering point sources in heat and mass transfer problems. [ABSTRACT FROM AUTHOR]

This paper aims at reviewing and analysing the method of reflections, which is an iterative procedure designed for solving linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on boundary integral representation. For both the sequential and parallel forms of the method appearing in the literature, we interpret the procedure in terms of projection operators. Using a Hilbert space setting and orthogonality, we prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. Several examples of boundary value problems that enter such a framework are given, an alternative proof of convergence is provided in a case which does not. A few numerical tests conclude the study. [ABSTRACT FROM AUTHOR]

A class of two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ (1, 2] with a convection term is considered. Its boundary conditions are of Robin type including the Dirichlet boundary conditions as a special case. An explicit formula for the associated Green function is obtained in terms of two-parameter Mittag-Leffler functions. This work improves Meng and Stynes' work [9] in three aspects. Firstly, the Green function is constructed by the use of the Laplace transform. This method may be more straightforward and easy to be generalized to solving other problems. Secondly, the monotonicity of a function is proven using monotone definition rather than using its derivative. Thirdly, a direct proof of the necessity of the positive property of Green function is given so that we can avoid the difficulty of construction any counter-example. [ABSTRACT FROM AUTHOR]

A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation. The idea gives a design of a special heat flux on the surface of the body such that from the corresponding temperature field on the surface one can extract the smallest radius of the sphere centered at an arbitrary given point in the whole space and enclosing unknown inclusions. Unlike before, the designed flux is free from a large parameter. An application of the idea to a coupled system of the elastic wave and heat equations are also given. [ABSTRACT FROM AUTHOR]

In [Nkemzi and Jung, 2013] explicit extraction formulas for the computation of the edge flux intensity functions for the Laplacian at axisymmetric edges are presented. The present paper proposes a new adaptation for the Fourier-finite-element method for efficient numerical treatment of boundary value problems for the Poisson equation in axisymmetric domains Ω̂ ⊂ ℝ3 with edges. The novelty of the method is the use of the explicit extraction formulas for the edge flux intensity functions to define a postprocessing procedure of the finite element solutions of the reduced boundary value problems on the two-dimensional meridian of Ω̂. A priori error estimates show that the postprocessing finite element strategy exhibits optimal rate of convergence on regular meshes. Numerical experiments that validate the theoretical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple positive solutions for boundary value problems of high-order Riemann–Liouville fractional differential equations involving the p-Laplacian operator. Not only new existence conclusions of two positive solutions are obtained by employing functional-type cone expansion-compression fixed point theorem, but also some sufficient conditions for existence of at least three positive solutions are established by applying the Leggett–Williams fixed point theorem. In addition, we demonstrate the effectiveness of the main result by using an example. [ABSTRACT FROM AUTHOR]

Existence results for the three-point fractional boundary value problem D α x (t) = f (t , x (t) , D α − 1 x (t)) , 0 < t < 1 , x (0) = A , x (η) − x (1) = (η − 1) B , are presented, where A , B ∈ R , 0 < η < 1 , 1 < α ≤ 2 . D α x (t) is the conformable fractional derivative, and f : [ 0 , 1 ] × R 2 → R is continuous. The analysis is based on the nonlinear alternative of Leray–Schauder. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a class of second order coefficient operators differential equation with general (possibly non local) boundary conditions. We obtain new results extending those given in a previous paper . Existence, uniqueness and optimal regularity of the strict solution are proved in UMD spaces, using the well‐known Dore–Venni theorem. [ABSTRACT FROM AUTHOR]

We consider boundary value problems for the Poisson equation on polygonal domains with general nonhomogeneous mixed boundary conditions and derive, on the one hand, explicit extraction formulas for the coefficients of the singularities. On the other hand, the formulas are used to construct efficient adaptations for the h -, p - and h p -versions of the finite element method for the numerical treatment. A priori error estimates show that the h -version of the finite element algorithm exhibits the same rate of convergence as it is known for problems with smooth solutions. However, the principal results of the present work are the robust exponential convergence results for the p - and h p -versions of the finite element method on quasiuniform meshes. In fact, it is shown that if the input data (source term and boundary data) are piecewise analytic, then with appropriate choices of conforming finite element subspaces V N of dimension N ∈ N , the p - and h p -versions of the finite element algorithms on quasiuniform meshes yield approximate solutions u N to the exact solution u that satisfy the estimates ‖ u − u N ‖ H 1 ( Ω ) ≤ C 1 e − b 1 N 2 3 and ‖ u − u N ‖ H 1 ( Ω ) ≤ C 2 e − b 2 N 1 3 , respectively. Several numerical experiments are included to illustrate the practical effectiveness of the proposed algorithms. The results show that the theoretical error analyses are attained within the range of engineering accuracy. [ABSTRACT FROM AUTHOR]

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set. [ABSTRACT FROM AUTHOR]

We study a nonlinear elliptic problem with non-local boundary conditions and variable exponent. We prove an existence and uniqueness result of weak solution to this problem with general maximal monotone graphs. [ABSTRACT FROM PUBLISHER]

This paper describes different representations for solution operators of Laplacian boundary value problems on bounded regions in RN,N≥2 and in exterior regions when N=3. Null Dirichlet, Neumann and Robin boundary conditions are allowed and the results hold for weak solutions in relevant subspaces of Hilbert–Sobolev space associated with the problem. The solutions of these problems are shown to be strong limits of finite rank perturbations of the fundamental solution of the problem. For exterior regions these expressions generalize multipole expansions. [ABSTRACT FROM AUTHOR]

This paper deals with an inverse monopolar source problem for the Poisson equation, from interior measurements, whose motivation lies in the sea water intrusion phenomenon. Global logarithmic stability estimates of locations and intensities for monopolar sources in this equation are established. To do that, we make an appropriate choice of a test function allowing to show a stability estimate with respect to boundary conditions and then we use an observability inequality for Laplace equation to control it by the interior measurements. This reveals, on the one hand, the distance between the source and on the other hand the gap between the associated measures. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system Here $${S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)$$ is the k -Hessian operator, a , p , f , a , p and f are continuous functions. [ABSTRACT FROM AUTHOR]

In this article, we consider the following boundary value problem of nonlinear fractional differential equation with p-Laplacian operator: where $1<\alpha\leq2$ is a real number, $D^{\alpha}$ is the conformable fractional derivative, $\phi_{p}(s)=\vert s\vert ^{p-2}s$ , $p>1$ , $\phi_{p}^{-1}=\phi_{q}$ , $1/p+1/q=1$ , and $f:[0, 1]\times[0,+\infty)\to[0,+\infty)$ is continuous. One of the difficulties here is that the corresponding Green's function $G(t, s)$ is singular at $s= 0$ . By the use of an approximation method and fixed point theorems on cone, some existence and multiplicity results of positive solutions are acquired. Some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

We consider the eigenvalue problem with Robin boundary condition Δu+λu= 0 inΩ,∂u/∂ν+αu= 0 on∂Ω, whereΩ ⊂ℝn,n ≥2 is a bounded domain with a smooth boundary,νis the outward unit normal,αis a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem forα →+∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions. [ABSTRACT FROM PUBLISHER]

In this paper, we give three kinds of Almansi representations for super-polyharmonic functions. As applications of these representations, we investigate two kinds of boundary value problems in superspace and construct solutions to the super-Poisson equation and the inhomogeneous super-polyharmonic equation. [ABSTRACT FROM PUBLISHER]

We consider the Laplacian on a class of smooth domains Ω ⊂ R ν , ν ≥ 2 , with attractive Robin boundary conditions: Q α Ω u = − Δ u , ∂ u ∂ n = α u on ∂ Ω , α > 0 , where n is the outer unit normal, and study the asymptotics of its eigenvalues E j ( Q α Ω ) as well as some other spectral properties for α tending to +∞. We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact C 2 boundaries we show that, for each fixed j , E j ( Q α Ω ) = − α 2 + μ j ( α ) + O ( log ⁡ α ) , where μ j ( α ) is the j th eigenvalue of the operator − Δ S − ( ν − 1 ) α H with ( − Δ S ) and H being respectively the positive Laplace–Beltrami operator and the mean curvature on ∂Ω. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues for non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian − Δ S − ( ν − 1 ) α H enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues for large α under various geometric assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of Q α Ω for large α . [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse scattering of tapered waves from sound soft surfaces or horizontally polarized electromagnetic waves from perfectly conducting surfaces. By the related Fréchet derivative with respect to a given surface of the solution of direct scattering problems, an efficient algorithm is proposed to reconstruct the parameters of unbounded rough surfaces from a set of scattered field measurements. Tapered wave is introduced to realize the asymptotic truncation for unbounded fractal rough surface. In order to improve the results of reconstruction, multi-angle and multi-frequency incidence strategy has been used. The numerical results show that the method yields satisfactory reconstructions for some rough surface profiles. [ABSTRACT FROM PUBLISHER]

We consider a model for the spreading of a substance through an incompressible fluid in a perforated domain,with. The fluid flows in a domain containing a periodical set of perforations () placed along an inner surface. The size of the perforations is much smaller than the size of the characteristic period. An adsorption phenomena can occur on the boundaries of the perforations, where we assume a strongly nonlinear adsorption law with a large adsorption parameter. An advection term appears in the partial differential equation. We obtain the homogenized model which also involves a nonlinear transmission condition for the normal derivative on. The ‘strange term’ arising in this transmission condition is a nonlinear function implicitly defined by a functional equation. We deal with critical relations both for the size of perforations and the adsorption parameter while we use the energy method for variational inequalities to show the convergence. [ABSTRACT FROM PUBLISHER]

An augmented Lagrangian method, based on boundary variational formulations and fixed point method, is designed and analyzed for the Signorini problem of the Laplacian. Using the equivalence between Signorini boundary conditions and a fixed-point problem, we develop a new iterative algorithm that formulates the Signorini problem as a sequence of corresponding variational equations with the Steklov-Poincaré operator. Both theoretical results and numerical experiments show that the method presented is efficient. [ABSTRACT FROM AUTHOR]

The elliptic and the parabolic equations with Dirichlet boundary conditions in fractured media are considered. The fractured media consist of a periodic connected high permeability sub-region and a periodic disconnected matrix block subset with low permeability. Let ϵ ∈ ( 0 , 1 ] denote the size ratio of the matrix blocks to the whole domain and let ω 2 ∈ ( 0 , 1 ] denote the permeability ratio of the disconnected subset to the connected sub-region. It is proved that the W 1 , p norm of the elliptic and the parabolic solutions in the high permeability sub-region are bounded uniformly in ω , ϵ . However, the W 1 , p norm of the solutions in the low permeability subset may not be bounded uniformly in ω , ϵ . For the elliptic and the parabolic equations in periodic perforated domains, it is also shown that the W 1 , p norm of their solutions are bounded uniformly in ϵ . [ABSTRACT FROM AUTHOR]

In this paper, we prove that a bifurcation phenomenon exists in a one-phase singularly perturbed free boundary problem of phase transition. Namely, the uniqueness of a solution of the one-phase problem breaks down as the boundary data decreases through a threshold value. The minimizer of the functional in consideration separates from the trivial harmonic solution. Moreover, we prove a third solution, a critical point of the functional being minimized, exists in this case by using the Mountain Pass Lemma. We prove convergence of the evolution with initial data near the minimizer or the trivial harmonic solution to the minimizer or to the trivial solution respectively, which means both the minimizer and the trivial harmonic solution are stable, while a saddle point solution of the free boundary problem is unstable in this sense. [ABSTRACT FROM AUTHOR]

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $$\mathbb {R}^n$$ - the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain $$\Omega $$ is $$C^2$$ , we prove a doubling property for the eigenfunction $$u$$ . We estimate the Hausdorff $$\mathcal H^{n-2}$$ -measure of the nodal set of $$u|_{\partial \Omega }$$ in terms of the eigenvalue $$\lambda $$ as $$\lambda $$ grows to infinity. In case that the domain $$\Omega $$ is analytic, we prove a polynomial bound O( $$\lambda ^6$$ ). Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous works [Garofalo and Lin, Commun Pure Appl Math 40(3):347-366, ; Lin, Commun Pure Appl Math 44(3):287-308, ]. [ABSTRACT FROM AUTHOR]

We consider the Laplacian with attractive Robin boundary conditions, in a class of bounded smooth domains $$\Omega \in \mathbb {R}^\nu $$ ; here $$n$$ is the outward unit normal and $$\alpha >0$$ is a constant. We show that for each $$j\in \mathbb {N}$$ and $$\alpha \rightarrow +\infty $$ , the $$j$$ th eigenvalue $$E_j(Q^\Omega _\alpha )$$ has the asymptotics where $$H_\mathrm {max}(\Omega )$$ is the maximum mean curvature at $$\partial \Omega $$ . The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $$H_\mathrm {max}$$ . In particular, we show that the ball is the strict minimizer of $$H_\mathrm {max}$$ among the smooth star-shaped domains of a given volume, which leads to the following result: if $$B$$ is a ball and $$\Omega $$ is any other star-shaped smooth domain of the same volume, then for any fixed $$j\in \mathbb {N}$$ we have $$E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )$$ for large $$\alpha $$ . An open question concerning a larger class of domains is formulated. [ABSTRACT FROM AUTHOR]

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78-95, ) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier-Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem, ) and study numerically the minimization problem at low and mid-high frequencies. Numerical examples in some relevant cases are also shown. [ABSTRACT FROM AUTHOR]

We consider a domain optimization problem of an unbounded domain, which models scattering of a time-harmonic acoustic wave at the junction of two closed waveguides. Solutions of our problem fulfill the Helmholtz equation with a real wavenumber, a modal radiation condition and homogeneous Dirichlet boundary conditions. We derive an a-priori bound for the solution on a certain class of domains (which is compact in the Hausdorff metric topology) and show that within this class the solution depends $$H^1$$ -continuously on the domain. Furthermore, we show some numerical examples to illustrate our results, which were calculated using the domain (or shape) derivative of our problem. [ABSTRACT FROM AUTHOR]

The incorporation of analytical kernel information is exploited in the construction of Nyström discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and fine grids, high-order polynomial interpolation, product integration performed on the fly, and iterative solution are some of the numerical techniques used to seek rapid and stable convergence of computed fields in the entire computational domain. [ABSTRACT FROM AUTHOR]

We study the size of nodal sets of Laplacian eigenfunctions on compact Riemannian manifolds without boundary and recover the currently optimal lower bound by comparing the heat flow of the eigenfunction with that of an artificially constructed diffusion process. The same method should apply to a number of other questions; we use it to prove a sharp result saying that a nodal domain cannot be entirely contained in a small neighborhood of a “reasonably flat” surface and recover an older result of Cheng. The arising concepts can be expected to have many more connections to classical theory and we pose some conjectures in that direction. [ABSTRACT FROM PUBLISHER]

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self-adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self-adjoint operators. We then characterize the lower bounded self-adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions. [ABSTRACT FROM AUTHOR]

We study two Dirichlet boundary value problems involving the weak Laplacian on the Sierpiński gasket. Using variational methods we prove the existence and multiplicity of weak solutions for these problems. [ABSTRACT FROM PUBLISHER]

We study the Neumann problem for the dissipative Helmholtz equation in the exterior of open Lipschitz surfaces inR3. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of a double layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable. [ABSTRACT FROM AUTHOR]

We consider the reconstruction of an obstacle with impedance boundary in an inverse scattering problem from far-field data. Both the boundary shape and the boundary impedance are unknown. By first transforming the far field into the near field, we consider this inverse problem as the detection of a void with impedance boundary embedded in bounded homogeneous media domainG. Then we set the Dirichlet-to-Neumann boundary condition in ∂Gand solve this interior boundary reconstruction problem in terms of the method of fundamental solution (MFS) with the near field on ∂Gas input inversion data. For this problem, both the locations of source points and the obstacle boundary as well as the expansion coefficients of wave field are taken to be the unknowns under the framework of regularizing optimization. The property of this optimization problem such as the existence and convergence of the minimizer is analysed. Numerical examples are presented for several configurations to show the validity of our proposed scheme. [ABSTRACT FROM AUTHOR]

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces. [ABSTRACT FROM AUTHOR]

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace's equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O( h), where h is the mesh step, when the boundary functions are from C, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O( h(|ln h| + 1)), when the boundary functions are from C, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods. [ABSTRACT FROM AUTHOR]

Classes of problems of wave diffraction by a plane angular screen occupying an infinite 270 degrees wedge sector are studied in a Bessel potential spaces framework. The problems are subjected to different possible combinations of boundary conditions on the faces of the wedge. Namely, under consideration there will be boundary conditions of Dirichlet-Dirichlet, Neumann-Neumann, Neumann-Dirichlet, impedance-Dirichlet, and impedance-Neumann types. Existence and uniqueness results are proved for all these cases in the weak formulation. In addition, the solutions are provided within the spaces in consideration, and higher regularity of solutions are also obtained in a scale of Bessel potential spaces. [ABSTRACT FROM AUTHOR]

In this article, we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients, and inhomogeneities. The main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in Grothaus and Raskop (Stochastics 2006; 78(4):233-257). Moreover we prove regularization results for the weak solutions of both the inner and the outer problem. We investigate the nonadmissible direction for the oblique vector field, state results with stochastic inhomogeneities, and provide a Ritz-Galerkin approximation. The results are applicable to problems from Geomathematics (see, e.g., Freeden and Maier (Electronic Trans. Numer. Anal. 2002; 14:40-62 and Bauer, An Alternative Approach to the Oblique Derivative Problem in Potential Theory, Shaker Verlag, Aachen, 2004)). [ABSTRACT FROM AUTHOR]

Frequency sounding of layered media is modeled by a hyperbolic problem. Within the framework of this model, we formulate an inverse problem. Applying the Laplace transform and introducing the impedance function, the latter is first reduced to the inverse boundary value problem for the Riccati equation and then to the Cauchy problem for a first-order quadratic equation. The advantage of such transformations is that the quadratic equation does not contain an unknown coefficient. For a specific class of data, it is shown that the Cauchy problem is uniquely solvable. Based on the asymptotic behavior of solutions to both the Riccati and quadratic equations, a stable reconstruction algorithm is constructed. Its feasibility is demonstrated in computational experiments. [ABSTRACT FROM AUTHOR]

Let &Tscr; k be a sequence of triangulations of a polyhedron Ω &NotSubset; &reals; n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m . Let u k &Element; S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a positive solution to a boundary value problem of fractional differential equations with the p-Laplacian operator is studied. By applying a monotone iterative method, some existence results of positive solutions are obtained. In addition, an example is included to illustrate the main results. [ABSTRACT FROM AUTHOR]