Adaptive and parallel algorithms for solving partial differential equations with variable coefficients on sparse grids
Final Report Abstract
Finite element methods are used to solve partial differential equations numerically. They are characterized by high accuracy and accurate convergence theory. However, they can hardly be used efficiently for partial differential equations of higher dimensions d>3, since the computational effort increases with the order O(Nd), where N is the number of unknowns in one spatial direction. Furthermore, classic parallelization techniques, such as domain decomposition, do not allow a reduction in the computational effort, since they are inefficient at higher dimensions. Theoretically, a hierarchical approach can significantly reduce the computational effort in higher dimensions. This approach is called sparse grids. It leads to only ����(���� log(����)����−1 ) grid points compared to O(Nd) grid points of classic full finite element grids. However, it is not trivial to develop algorithms, which numerically solve partial differential equations with a computational effort of the same order. In previous scientific work, algorithms were developed that can only be used in case of simple coefficients in the differential equation or in case of non-adaptive grids. In this research project, novel algorithms were developed that enable a much wider range of applications. On the one hand, locally adaptive thin grids could be created, which also allow discretization of partial differential equations with variable coefficients. To this end, locally adaptive sparse grids were developed, which also allow discretization of partial differential equations with variable coefficients. The concept of this discretization extends the discretization of semi-orthogonality with prewavelets as hierarchical basis functions in a suitable way. The discretization of general variable coefficients is particularly difficult. In previous work, only variable coefficients that have a tensor product structure were considered in numerical simulations. This makes both the storage and the calculation of the local stiffness matrices much easier than with general variable coefficients. The reason for the high computational effort for general coefficients is the anisotropy of the geometry of local stiffness elements and that the resulting local stiffness matrix has O((2d)2) = O(4d) entries. These problems were solved by a suitable Monte Carlo calculation of the local stiffness matrices. In order to obtain simulation results even in case of high dimensions d=4-6, suitable parallelization techniques for high-performance computers were developed. Simulation results show good convergence of the new finite element method even in case of high dimensions d = 4-6, in case of difficult variable coefficients with singularities and in case of solutions that require locally refined adaptive grids.
Link to the final report
https://oa.tib.eu/renate/handle/123456789/23659
Publications
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Discretization of PDEs with variable coefficients using locally adaptive sparse grids. PAMM, 23(4).
Scherner‐Grießhammer, Riccarda & Pflaum, Christoph
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Solving PDEs With Variable Coefficients on Locally Adaptive Sparse Grids and Corresponding Refinement Strategies. Lecture Notes in Computational Science and Engineering, 350-359. Springer Nature Switzerland.
Scherner-Grießhammer, Riccarda & Pflaum, Christoph
