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Adaptive mesh refinement (AMR) changes the spacing of grid points, to change how accurately the solution is known in that region. In the shallow water example, the grid might in general be spaced every few feet—but it could be adaptively refined to have grid points every few inches in places where there are large waves.
Mesh generation is deceptively difficult: it is easy for humans to see how to create a mesh of a given object, but difficult to program a computer to make good decisions for arbitrary input a priori. There is an infinite variety of geometry found in nature and man-made objects. Many mesh generation researchers were first users of meshes.
Adaptive multigrid exhibits adaptive mesh refinement, that is, it adjusts the grid as the computation proceeds, in a manner dependent upon the computation itself. [18] The idea is to increase resolution of the grid only in regions of the solution where it is needed.
Mesh adaptation on the whole or parts of the geometry, for stationary, eigenvalue, and time-dependent simulations and by rebuilding the entire mesh or refining chosen mesh elements. conforming and non-conforming adaptive refinement for tensor product and simplex meshes Only h h-, p-, and hp-adaptivity for both continuous and discontinuous ...
The mesh is an integral part of the model and must be controlled carefully to give the best results. Generally, the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge, and further mesh refinement does not increase accuracy. [30]
hp-FEM is a generalization of the finite element method (FEM) for solving partial differential equations numerically based on piecewise-polynomial approximations. hp-FEM originates from the discovery by Barna A. Szabó and Ivo Babuška that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into ...
A coarse mesh may provide an accurate solution if the solution is a constant, so the precision depends on the particular problem instance. One can selectively refine the mesh in areas where the solution gradients are high, thus increasing fidelity there. Accuracy, including interpolated values within an element, depends on the element type and ...
Plate tectonics is a theory suggesting that the Earth's lithosphere is essentially composed of plates floating on the mantle. [97] The mantle convection model is fundamental in modeling the plates floating on it, and there are two major approaches to incorporate the plates into this model: rigid-block approach and rheological approach. [ 2 ]