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The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal.56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms. [ABSTRACT FROM AUTHOR]

This paper builds on the algebraic theory in the companion paper [S. F. McCormick, J. Benzaken, and R. Tamstorf, SIAM J. Sci. Comput., 43 (2021), pp. S392--S419] to obtain discretization-error-accurate solutions for linear elliptic partial differential equations (PDEs) by mixed-precision multigrid solvers. It is often assumed that the achievable accuracy is limited by discretization or algebraic errors. On the contrary, we show that the quantization error incurred by simply storing the matrix in any fixed precision quickly begins to dominate the total error as the discretization is refined. We extend the existing theory to account for these quantization errors and use the resulting bounds to guide the choice of four different precision levels in order to balance quantization, algebraic, and discretization errors in the progressive-precision scheme proposed in the companion paper. A remarkable result is that while iterative refinement is susceptible to quantization errors during the residual and update computations, the V-cycle used to compute the correction in each iteration is much more resilient and continues to work if the system matrices in the hierarchy become indefinite due to quantization. As a result, the V-cycle only requires relatively few bits of precision per level. Based on our findings, we outline a simple way to implement a progressive-precision full multigrid (FMG) solver with minimal overhead and demonstrate as an example that the one-dimensional biharmonic equation can be solved reliably to any desired accuracy using just a few V-cycles when the underlying smoother works well. Additionally, we show that the progressive-precision scheme leads to memory savings of up to 50\% compared to fixed precision. [ABSTRACT FROM AUTHOR]