The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device -which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved -independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local -making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.

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The finite mass method is a gridless Lagrangian method to simulate compressible flows that has been introduced in a recent paper from Gauger, Leinen, and Yserentant [SIAM J. Numer. Anal., 37 (2000), pp. 1768-1799. It is based on a discretization of mass, not of space as with classical discretization schemes. Mass is subdivided into little mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can change their shape to follow the motion of the fluid. Only free flows in vacuum have been considered so far. In this article, a concept is presented to extend the method to flows in domains having boundaries. It maintains the second order accuracy of the basic method and can be implemented along the same lines.

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The finite mass method is a gridless Lagrangian method to simulate compressible flows that has been developed by Gauger, Leinen, and Yserentant [SIAM J. Numer. Anal., 37 (2000), pp. 1768-1799. It is based on a discretization of mass, not of space as with classical discretization schemes. Mass is subdivided into little mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can change their size, orientation, and shape to follow the motion of the fluid. The ability of the mass packets to deform with the flow is the reason for the high accuracy of the finite mass method but also requires that the computation is newly set up from time to time. Such restart procedures are described in this article. Introduction. The finite mass method was introduced in [4] and [7] and is founded on concepts developed in [12], [13], and [14]. It is a gridless Lagrangian method to solve problems in fluid mechanics and is based, in contrast to standard methods, on a discretization of mass, not of space. Mass is subdivided into small mass packets that move independently of each other under the influence of internal and external forces and the laws of thermodynamics and normally overlap. The approximations the finite mass method delivers are smooth functions and not discrete measures, so that it comes closer to classical discretizations than to usual particle methods.The high accuracy of the finite mass method originates from the ability of the mass packets to rotate, to expand, to contract, and to deform with the flow, but this ability also leads to problems when particles degenerate or no longer align sufficiently well with each other. In such situations, the computation must be newly set up, and the mass density, the velocity field, and the other field quantities must be represented by a new set of nondeformed particles again aligned with an appropriate grid. At first sight, this reminds one of Godunov-type methods [8], [9], for example, in which the information is remapped after every timestep to the given grid. But in comparison with such grid-oriented methods, recovery steps of this kind are necessary only very seldomly in the finite mass method, typically after certain fixed time intervals, and not after every timestep. Therefore much less diffusion is introduced, and the method keeps its favorable properties as an essentially gridless Lagrangian method.The restart procedure developed in this article is of purely local nature and maintains the formal order of accuracy of the finite mass method. It is itself gridless, as it concerns the transfer of the velocity field to the new particles. Quantities such as the mass density are transferred by a quasi-interpolation method to the new representation. More involved total variation diminishing schemes related to those [1], [11]