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Some new analysis results for a class of interface problems. (English) Zbl 1338.65197

Authors’ abstract: Interface problems modeled by differential equations have many applications in mathematical biology, fluid mechanics, material sciences, and many other areas. Typically, interface problems are characterized by discontinuities in the coefficients and/or the Dirac delta function singularities in the source term. Because of these irregularities, solutions to the differential equations are not smooth or discontinuous. In this paper, some new results on the jump conditions of the solution across the interface are derived using the distribution theory and the theory of weak solutions. Some theoretical results on the boundary singularity in which the singular delta function is at the boundary are obtained. Finally, the proof of the convergence of the immersed boundary (IB) method is presented. The IB method is shown to be first-order convergent in \(L^{\infty}\) norm.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34A36 Discontinuous ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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