Abstract
As long as the solutions of boundary value problems are sufficiently regular, the refinement of the mesh size h and the increase of the approximation order k in the discretization space \(V_h^k\) yields an improvement in the accuracy. In particular, this yields optimal convergence rates. But, in most applications the regularity of the solution is restricted due to corners of the domain or jumping physical quantities. Therefore, it is essential to adapt the solution process to the underlying problem in order to retrieve optimal approximation properties. In this chapter, we deal with a posteriori error estimates which can be used to drive an adaptive mesh refinement procedure and we recover optimal rates of convergence for the adaptive methods in the numerical experiments in the presence of singularities. For the error estimation, we cover the classical residual based error estimator as well as goal-oriented techniques on general polytopal meshes. Whereas, we derive reliability and efficiency estimates for the first mentioned estimator, we discuss the benefits and potentials of the second one for general meshes.
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Weißer, S. (2019). Adaptive BEM-Based Finite Element Method. In: BEM-based Finite Element Approaches on Polytopal Meshes. Lecture Notes in Computational Science and Engineering, vol 130. Springer, Cham. https://doi.org/10.1007/978-3-030-20961-2_5
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