Adaptive finite element methods for differential equations

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These Lecture Notes are based on material presented by the second author during a lecture series at ETH Zurich in the summer term of 2002. They explore 'self adaptivity' in the numerical solution of differential equations, focusing on Galerkin finite element methods. Key topics include a posteriori error estimation and automatic mesh adaptation. In addition to traditional energy-norm error control, a novel duality-based technique known as the Dual Weighted Residual (DWR) method is discussed for goal-oriented error estimation. This method facilitates the efficient computation of various physical quantities by adapting the computational mesh appropriately, which is essential in technical application design cycles. For instance, it allows for the computation and minimization of the drag coefficient of a body in a viscous flow by adjusting control parameters, followed by stability analysis through eigenvalue problems. 'Goal-oriented' adaptivity aims to achieve these objectives cost-effectively. The foundational aspects of the DWR method and its applications are elaborated in survey articles, including works by R. Rannacher and M. Braack, which address error control in finite element computations and adaptive methods for low Mach-number flows with chemical reactions.