Abstract
The convergence analysis with rates for adaptive least-squares finite element methods (ALSFEMs) combines arguments from the a posteriori analysis of conforming and mixed finite element schemes. This paper provides an overview of the key arguments for the verification of the axioms of adaptivity for an ALSFEM for the solution of a linear model problem. The formulation at hand allows for the simultaneous analysis of first-order systems of the Poisson model problem, the Stokes equations, and the linear elasticity equations. Following [Carstensen and Park, SIAM J. Numer. Anal. 53(1), 2015], the adaptive algorithm is driven by an alternative residual-based error estimator with exact solve and includes a separate marking strategy for quasi-optimal data resolution of the right-hand side. This presentation covers conforming discretisations for an arbitrary polynomial degree and mixed homogeneous boundary conditions.
Funding statement: This work was supported by the Deutsche Forschungsgemeinschaft Priority Program 1748 ‘Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis’ within the project ‘Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics’ (CA 151/22-1 and CA 151/22-2), by the Berlin Mathematical School, and by the Austrian Science Fund (FWF) through the project ‘Computational nonlinear PDEs’ (grant P33216).
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