Boundary element method application to numerical solving of linear boundary-value problems in domains with strongly segmented boundary

В настоящей работе метод граничных элементов был применен к решению краевых задач для уравнения Лапласа в плоской области с сильно сегментированной границей. Особое внимание было уделено точности численного решения, которая исследовалась путем численного эксперимента на специально подобранных тестовых задачах, имеющих аналитические решения в квадратурах. Было реализовано два алгоритма метода граничных элементов: традиционный с решением системы линейных алгебраических уравнений методами гауссовского исключения, и итерационный, при этом в итерационном алгоритме использовались функции Грина или их вычислительные аналоги. Результаты работы могут быть использованы при создании специализированного программного обеспечения соответствующего назначения.One of the most serious problems of modern numerical analysis is boundary-value problem solution in domains of complex geometrical shapes. Such problems are proved especially difficult for the domains with strongly segmented boundary, which meansthat the boundary is divided into isolated pieces. Such situations are specific for heterogeneous media. In such situations local approximation methods have to deal with the insuperable difficulties such as constructing computational grid and subsequent solving rather sophisticated systems of linear algebraic equations. The methods of global approximations and, first of all, methods of computational potential theory do not have similar difficulties, nevertheless they have to overcome a lot of problems. Boundary element method is applied in thepresent work to solve boundary-value problems for Laplace equations in plane domain with strongly segmented boundary. Special attention in the work was paid to accuracy of numerical solutions. The accuracy is investigated by a numerical experiment using specially selected test problems, which have the known analytical solutions in quadrature. Two boundary element algorithms are implemented. The first one is the traditional approach with Gauss elimination algorithm for solving linear algebraic equation system. The second one is an iterative approach with possible using of Green’s functions or their computational analogs in the iterative procedure. The results obtained in the work can be applied for creating specialized software of corresponding purposes

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We analyze firms’ incentives to coordinate on the introduction of a sustainable product variant when consumers’ preferences for sustainability depend on the perceived social norm, which in turn is shaped by average consumption behavior.We show that such preferences could lead to multiple equilibria. If the level of competition among potential adopters is very low and adoption of the sustainable variant allows them to sufficiently expand their aggregate market share, they will coordinate on introducing the sustainable variant when a lenient legal regime makes this feasible. If competition among them is intense and market expansion under the sustainable variant is very limited, coordination can forestall the adoption of the sustainable variant. Our analysis thus both confirms and qualifies the notion of a sustainability “first-mover disadvantage” as a justification for an agreement between competitors, which has gained traction in antitrust.We also provide empirical evidence for norm-based sustainability preferences

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We explore why venture capital funds limit the amount of capital they raise and do not reinvest the proceeds. This structure is puzzling because it leads to a succession of several funds financing each new venture which multiplies the well known agency problems. We argue that an inside investor cannot provide a hard budget constraint while a less well informed outsider can. Therefore, the venture capitalist delegates the continuation decision to the outsider by ex ante restricting the amount of capital he has under management. The soft budget constraint problem becomes the more important the higher the entrepreneur’s private benefits are and the higher the probability of failure of a project is

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