The DWR method is a very successful approach for goal-oriented a posteriori error estimation in finite element analysis. We discuss the underlying technique, for which we refer to the early surveys [1, 2]. Using basic mathematical results like the fundamental theorem of calculus, we obtain an identity to represent the error in a user defined quantity of interest by residuals. However, the identity cannot be evaluated numerically. Hence, we have to approximate it by appropriate techniques. One main mathematical challenge in this context is to prove that the approximations lead to an asymptotic exact error estimator. We review first approaches using high regularity demands and discuss new results from [3], which are based on more realistic assumptions.
The only assumption in the theoretical derivation of the DWR method is that the differential operator is at least twice or preferable three times directional differentiable. In contact problems, the differential operator is usually not directional differentiable in the classic sense at all. Hence, the DWR approach cannot be applied directly. We present a modification of the method such that the main properties are kept and it can be applied on contact problems, cf. [4, 5].
References
[1] W. Bangerth and R. Rannacher: Adaptive finite element methods for differential equations. Lectures in Mathematics, ETH Zürich, Birkhäuser, Basel, 2003.
[2] R. Becker and R. Rannacher: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica, 10, 1-102, 2001.
[3] B. Endtmayer, U. Langer, and T. Wick: Two-side a posteriori error estimates for the DWR method. SIAM Journal of Scientific Computing, 42(1), A371-A394, 2020.
[4] A. Rademacher: NCP-function based dual weighted residual error estimators for Signorini's problem. SIAM Journal of Scientific Computing, 38(3):
A1743-A1769, 2016.
[5] A. Rademacher: Mesh and model adaptivity for frictional contact problems. Numerische Mathematik, 142(3): 465-523, 2019.