Book ChapterDOI
Construction and Convergence of Difference Schemes for a Model Elliptic Equation with Dirac-delta Function Coefficient
Boško S. Jovanović,Juri D. Kandilarov,Lubin G. Vulkov +2 more
- pp 431-438
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TLDR
Estimates for the rate of convergence in discrete energetic Sobolev's norms compatible with the smoothness of the solution are presented.Abstract:
We first discuss the difficulties that arise at the construction of difference schemes on uniform meshes for a specific elliptic interface problem. Estimates for the rate of convergence in discrete energetic Sobolev's norms compatible with the smoothness of the solution are also presented.read more
Citations
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Journal ArticleDOI
The immersed interface method for two-dimensional heat-diffusion equations with singular own sources
TL;DR: In this article, the authors developed the immersed interface method (IIM) due to [R.J. LeVeque, Z. Li] for solving parabolic equations with singular own sources.
Book ChapterDOI
Finite Difference Methods for Boundary Value Problems
Uri M. Ascher,Linda R. Petzold +1 more
Journal ArticleDOI
On the Convergence of Difference Schemes for Hyperbolic Problems with Concentrated Data
TL;DR: For finite difference schemes approximating several one-dimensional initial-boundary value problems convergence rate estimates in special discrete energetic Sobolev's norms, compatible with the smoothness of the solutions, are obtained.
Journal ArticleDOI
Analysis of Immersed Interface Difference Schemes for Reaction-diffusion Problems with Singular Own Sources
TL;DR: In this article, the authors analyzed immersed interface difference schemes for onedimensional reaction-diffusion equations with singular own linear and nonlinear sources and derived error bounds in the infinity norm based on the maximum principle.
Journal ArticleDOI
Finite Difference Schemes for Partial Differential Equations with Weak Solutions and Irregular Coefficients
TL;DR: In this paper, a survey of the results concerning the convergence of difference schemes for boundary value problems with generalized solutions from the Sobolev space is presented, in particular for some problems with singular coefficients.
References
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Book
The Theory of Difference Schemes
TL;DR: The theory of difference schemes was introduced in this paper, where homogeneous difference schemes and different schemes for elliptic and time-dependent equations with constant coefficients were studied. And the theory of differenceschemes symbols was discussed.
Journal ArticleDOI
Analysis of a one-dimensional model for the immersed boundary method
R. P. Beyer,Randall J. LeVeque +1 more
TL;DR: In this paper, the authors studied numerical methods for the one-dimensional heat equation with a singular forcing term, where the delta function was replaced by a discrete approximation, and the resulting equation was solved by a Crank-Nicolson method on a uniform grid.
Dissertation
The immersed interface method: a numerical approach for partial differential equations with interfaces
Journal ArticleDOI
The Immersed Interface Method for Nonlinear Differential Equations with Discontinuous Coefficients and Singular Sources
Andreas Wiegmann,Kenneth P. Bube +1 more
TL;DR: In this paper, the authors extend the immersed interface method of LeVeque and Li to find numerical solutions of one-dimensional parabolic partial differential equations of the form $u_t = (β(x,t) u_x)