A representation of the solution to an ellipticboundaryvalue problem in the vicinity of a corner point on the discontinuity line of the coefficient of the higher order derivative is constructed. The study is based o...
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A representation of the solution to an ellipticboundaryvalue problem in the vicinity of a corner point on the discontinuity line of the coefficient of the higher order derivative is constructed. The study is based on the method of additive separation of singularities proposed by Kondrat'ev.
The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither H2\documentclass[12pt]{minimal} \usepackage{amsma...
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The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>2$$\end{document}-regularity nor L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L<^>{\infty } $$\end{document}-error estimation, but only H01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H<^>1_0 $$\end{document}-error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require L infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L<^>{\infty } $$\end{document}-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.
Given an isometric action of a discretegroup G on a compact manifoldM with boundary and a G-invariant ellipticboundaryvalue problem D on M, we consider its twisting by projections over the crossed product algebra C-...
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Given an isometric action of a discretegroup G on a compact manifoldM with boundary and a G-invariant ellipticboundaryvalue problem D on M, we consider its twisting by projections over the crossed product algebra C-infinity( M) x G. The twisted problemis Fredholmand we compute its index in terms of the equivariant Chern character of the principal symbol of D and a noncommutative Chern character of P. In the special case, when D is the Dirichlet problem for the Euler operator, the index is expressed as a linear combination of the Euler characteristics of the fixed point submanifolds of the group action.
Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundaryvalue problem for a linear ellipti...
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Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundaryvalue problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in SobolevSlobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in SobolevSlobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (ha) of approximation as in the smooth case. (c) 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012
The main result of the paper is the proof of the applicability of hypercomplex methods for elliptic boundary value problems in outer domains, i.e. domains, which lies outside a closed compact surface.
The main result of the paper is the proof of the applicability of hypercomplex methods for elliptic boundary value problems in outer domains, i.e. domains, which lies outside a closed compact surface.
This paper studies the regularity of solutions to boundaryvalueproblems for the Laplace operator on Lipschitz domains Omega in R(d) and its relationship with adaptive and other nonlinear methods for approximating th...
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This paper studies the regularity of solutions to boundaryvalueproblems for the Laplace operator on Lipschitz domains Omega in R(d) and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in L(p)(Omega) are the Besov spaces B-tau(alpha)(L(tau)(Omega)), tau := (alpha/d + 1/p)(-1). Thus, the regularity of the solution in this scale of Besov spaces is investigated with the aim of determining the largest a for which the solution is in B-tau(alpha)(L tau(Omega)). The regularity theorems given in this paper build upon the recent results of Jerison and Kenig [10]. The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions.
The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundaryvalue problem onto a fix...
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The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundaryvalue problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.
The paper proves convergence of unsymmetric radial basis functions (RBFs) collocation for second order elliptic boundary value problems on the bounded domains. By using Schaback's linear discretization theory, L-2...
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The paper proves convergence of unsymmetric radial basis functions (RBFs) collocation for second order elliptic boundary value problems on the bounded domains. By using Schaback's linear discretization theory, L-2 error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. The present theory covers a wide range of kernel-based trial spaces including stationary and non-stationary approximation. The convergence rates depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. Some numerical examples are added for illustration.
This paper investigates the behavior of variational solutions of second-order elliptic mixed boundaryvalueproblems (MBVP) with real coefficients in n-dimensional domains with edges near the points where the edges ar...
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This paper investigates the behavior of variational solutions of second-order elliptic mixed boundaryvalueproblems (MBVP) with real coefficients in n-dimensional domains with edges near the points where the edges are vanishing. It is shown that the first coefficient involved in the decomposition of the solution into regular and singular part can be extended continuously in appropriate spaces across such points, thus showing that the standard decomposition formula holds also in such domains.
We consider a class of non-selfadioint elliptic boundary value problems on a bounded domain. Combining results on weighted eigenvalueproblems for non-selfadjoint operators with a general form of the method of sub-sup...
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