Maximum Principles for Fractional Diffusion Problems

doi:10.3390/sym18020272

Maximum Principles for Fractional Diffusion Problems

2026-01-31

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DOI：10.3390/sym18020272

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The maximum principle is a widely used qualitative property of linear (and not only) elliptic boundary value problems. A natural goal for developing numerical methods is for the approximate solution to have a similar property. In this case, we say that a discrete maximum principle holds. In many cases, such a requirement is critical to ensuring the reliability of computational models. Here, we consider multidimensional linear elliptic problems with diffusion and reaction terms. Such problems have been studied and analyzed for many decades. Since relatively recently, scientists have faced conceptually new challenges when considering anomalous (fractional) diffusion. In the present paper, we concentrate on the case of spectral fractional diffusion. Discretization was carried out using the finite difference method and the finite element method with a lumped mass matrix. In large-scale multidimensional problems, the computational complexity of dense matrix operations is critical. To overcome this problem, BURA (best uniform rational approximation) methods were applied to find the efficient numerical solutions of emerging dense linear systems. Thus, along with the need to satisfy the discrete maximum principle associated with the mesh method applied for discretization of the differential operator, the issue of the monotonicity of BURA numerical solution arises. The presented results are three-fold and include the following: (i) maximum principles for fractional diffusion-reaction problems; (ii) sufficient conditions for discrete maximum principles; and (iii) sufficient conditions for monotonicity of the investigated BURA- or BURA-like approximation methods. A novel, systematic theoretical analysis is developed for sub-diffusion with a fractional power alpha is an element of(1/2,1) and a constant reaction coefficient. The theoretical findings are further supported by numerical examples.

Keywords:

maximum principle

non-negativity preserving

fractional diffusion

rational approximation

Maximum Principles for Fractional Diffusion Problems

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