In this work we have developed, implemented and tested a new multilevel hybrid algorithm for the adaptive finite element solution of a general class of variational problems. Our multilevel hybrid algorithm is a combination of node movement, edge swapping and local h-refinement. The adaptive strategy used in our hybrid algorithm is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and the connectivity of the nodes is optimised. The grid is then locally refined and the new mesh is optimised in the same manner. Our hybrid algorithm does not need any global solution of the problem, it uses only local information to update the nodal solution values by solving the local variational problems on a relatively small domain with only few unknowns.\ud \ud The node movement strategy is based upon knowledge of a steepest descent direction for each node found by a gradient calculation. A derivation of the gradient of stored energy with respect to the position of nodes is provided. A strategy for the movement of interior as well as boundary nodes is then given. Edge/face swapping in two and three space dimensions is explained and algorithms for node movement and edge swapping are given. Detailed descriptions of the possible local refinement strategies in two and three space dimensions are provided. Possible variants of our hybrid algorithm are considered and aspects of our hybrid algorithm regarding the quality of the meshes achieved and the computational work undertaken are discussed with some preliminary results.\ud \ud We have applied our hybrid algorithm on a number of test problems: considering linear, nonlinear and system of equations in two and three space dimensions. A detailed comparison of the results produced by our hybrid algorithm with other adaptive approaches has been made for all of our test problems. Results presented indicate that our hybrid algorithm can produce better meshes, in both two and three space dimensions, than is possible by more conventional adaptive strategies
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