Abstract
In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests—with respect to accuracy and convergence—for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.
Highlights
Designing methods for the high-order accurate numerical approximation of partial differential equations (PDE) posed on composite domains with interfaces, or on irregular and geometrically complex domains, is crucial in the modeling and analysis of problems from science and engineering
There is extensive existing work addressing numerical approximation of PDE posed on composite domains with interfaces or irregular domains, for example, the boundary integral method [11,56], difference potentials method [3,6,26,27,58,65], immersed boundary method [30,42,61,74], immersed interface method [2,46,47,49,69], ghost fluid method [31,32,50,51], the matched interface and boundary method [82,84,85,86], Cartesian grid embedded boundary method [19,41,57,83], multigrid method for elliptic problems with discontinuous coefficients on an arbitrary interface [18], virtual node method [9,39], Voronoi interface method [35,36], the finite difference method [8,10,24,75,78,79] and finite volume method [22,34] based on mapped grids, or cut finite element method [13,14,15,37,38,71,76]
We propose a novel feature of Difference Potentials Method (DPM), extending the method originally developed in [67] and [3,4,5,6,26,28] to the composite domain problem (3–9) with implicitly-defined geometry
Summary
Designing methods for the high-order accurate numerical approximation of partial differential equations (PDE) posed on composite domains with interfaces, or on irregular and geometrically complex domains, is crucial in the modeling and analysis of problems from science and engineering. Such problems may arise, for example, in materials science (models for the evolution of grain boundaries in polycrystalline materials), fluid dynamics (the simulation of homogeneous or multi-phase fluids), engineering (wave propagation in an irregular medium or a composite medium with different material properties), biology (models of blood flow or the cardiac action potential), etc. It is still a challenge to design high-order accurate and computationally-efficient methods for PDE posed in these complicated geometries, especially for time-dependent problems, problems with variable coefficients, or problems with general boundary/interface conditions
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