The Convergence of the Finite Mass Method for Flows in Given Force and Velocity Fields
The finite mass method is a new Lagrangian method to solve problems in continuum mechanics, primarily to simulate compressible flows. It is directly founded on a discretization of mass, not of space as with classical discretization schemes. Mass is subdiv
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Introduction
The finite mass method was introduced in [3] and is based on concepts that have been developed in [5], [6], [7]. It is a gridless Lagrangian method to solve problems in continuum mechanics and is, in contrast to finite element and finite volume methods, directly based on a discretization of mass, not of space. Mass is subdivided into small mass packets each of which is equipped with finitely many internal degrees of freedom. These mass packets can move independently of each other, can rotate, expand and contract, and can even change their shape to follow the deformation of the material. The approximations the finite mass method produces are differentiable functions and not discrete measures such that in some way it comes much closer to classical discretizations schemes than to the usual Lagrangian particle methods. In [3], the finite mass method has been used to simulate free gas flows, as they are usually described by the compressible Euler and Navier-Stokes equations. Because of our lack of knowledge on the existence, uniqueness, and regularity of solutions of these equations, it is very hard to prove convergence for this situation. So it is already surprising that, analogously to the considerations in [5] and [7], the existence of limits satisfying the basic physical principles underlying gas dynamics can be shown. Therefore in this note we restrict our attention to two much simpler cases, the motion of mass under the influence of an external force field and the transport of mass in a given velocity field. For these cases, a much more detailed convergence analysis is possible, and in fact, one yields second order convergence as one could expect from the construction of the method. Thus our results also demonstrate that the ansatz functions used in the finite mass method are principally capable of producing second order approximations. M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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H. Y serentant
The method of proof is basically simple. As the mass packets do not interact under the given circumstances, the motion of each packet can be analyzed separately using techniques from the theory of ordinary differential equations and local linearizations of the fields under consideration. The resulting local maximum norm error estimates are then assembled to global estimates for the velocity field and the flow itself using the partition of unity given by the mass fractions of the single packets. These mass fractions already play a dominant role in the construction of the method and are used to define the approximate velocity, for example. In this respect our technique of proof therefore resembles the techniques applied in the partition of unity finite element method of Babuska and Melenk [1] or in the recent paper of Griebel and Schweitzer [4], for example. The argumentation for the mass density is quite different from the argumentation for the velocity field or the flow and is based on the fact that the discrete mass density
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