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Alves, Carlos J. S.; Martins, Nuno F. M.; Silvestre, Ana L.

Numerical methods with particular solutions for nonhomogeneous Stokes and Brinkman systems. (English) Zbl 1497.35149

Adv. Comput. Math. 48, No. 4, Paper No. 44, 23 p. (2022).
Summary: This paper deals with the numerical approximation of solutions of Stokes and Brinkman systems using meshless methods. The aim is to solve a problem containing a nonzero body force, starting from the well known decomposition in terms of a particular solution and the solution of a homogeneous force problem. We propose two methods for the numerical construction of a particular solution. One method is based on the Neuber-Papkovich potentials, which we extend to nonhomogeneous Brinkman problems. A second method relies on a Helmholtz-type decomposition for the body force and enables the construction of divergence-free basis functions. Such basis functions are obtained from Hänkel functions and justified by new density results for the space \(H^1(\Omega)\). Several 2D numerical experiments are presented in order to discuss the feasibility and accuracy of both methods.

Cited in 1 Document

MSC:

35J25 Boundary value problems for second-order elliptic equations
35Q35 PDEs in connection with fluid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs
65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Keywords:

nonhomogeneous Brinkman equations; nonhomogeneous Stokes equations; numerical approximation; meshfree method
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References:

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