Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

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The effects of insufficient and optimal sulfur (S) and nitrogen (N) supply on plant growth and glucosinolate formation were studied under controlled experimental conditions in broccoli "Monaco". Here, we report on the interaction between S and N supply, plant growth, and quality parameters and discuss the relevance of this interaction in relation to crop-management strategies. Broccoli plants supplied with insufficient amounts of S or N showed typical deficiency symptoms and yield decreases. In contrast, total glucosinolate concentrations were high at insufficient N supply, independent of the S level, and low at insufficient S supply in combination with an optimal N supply. This was mainly due to the presence of the alkyl glucosinolates glucoraphanin and glucoiberin. Furthermore, with S concentrations above 6 g (kg DM) -1 and an N : S ratio lower than 10:1, the glucosinolate concentrations were on average around 0.33 g (kg fresh matter) -1 and differed significantly from those plants characterized by an S concentration below 6 g (kg DM) -1 and an N : S ratio above 10:1. In addition, N : S ratios between 7:1 and 10:1 promoted plant yield and enhanced overall appearance. Therefore, to produce broccoli (and potentially other Brassicaceae) with higher crop yields and enhanced product quality in the field, it is vital to establish the optimal S and N nutritional status of the plant and to integrate this information into crop-management strategy programs.

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A multiresolution-based adaptation concept is proposed that aims at accelerating discontinuous Galerkin schemes applied to non-linear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. A central mathematical problem addressed in this work is then to show at least for scalar one-dimensional problems that choosing an appropriate threshold value, the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh. Numerical comparisons demonstrate the efficiency of the concept.

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