In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.
Application of nonlinearity continuation method to numerical solution of steady-state groundwater flow in variably saturated conditions is presented. In order to solve the system of nonlinear equations obtained by finite volume discretization of steady-state Richards equation, a series of problems with increasing nonlinearity are solved using the Newton method. This approach is compared to pseudo-transient method on several test cases, including real site problems and involving parallel computations.
Recently, the nonlinearity continuation method has been used to numerically solve boundary value problems for steady-state Richards equation. The method can be considered as a predictor-corrector procedure with the simplest form which has been applied to date having a trivial, zeroth-order predictor. In this article, effect of a more sophisticated predictor technique is examined. Numerical experiments are performed with finite volume and mimetic finite difference discretizations on various problems, including real-life examples.
Nonlinearity continuation method, applied to boundary value problems for steady-state Richards equation, gradually approaches the solution through a series of intermediate problems. Originally, the Newton method with simple line search algorithm was used to solve the intermediate problems. In this paper, other solvers such as Picard and mixed Picard-Newton methods are considered, combined with slightly modified line search approach. Numerical experiments are performed with advanced finite volume discretizations on model and real-life problems.
Oak Ridge National Lab, Climate Change Science Institute (CCSI), Environmental Sciences Division（橡树岭国家实验室气候变化科学研究所环境科学分部）；MATHICSE-CSQI, EPF de Lausanne（洛桑联邦理工学院数学与应用数学中心-复杂系统与量化不确定性研究中心）
We present a geometric variational discretization of nonlinear elasticity in 2D and 3D in the Lagrangian description. A main step in our construction is the definition of discrete deformation gradients and discrete Cauchy-Green deformation tensors, which allows for the development of a general discrete geometric setting for frame indifferent isotropic hyperelastic models. The resulting discrete framework is in perfect adequacy with the multisymplectic discretization of fluids proposed earlier by the authors. Thanks to the unified discrete setting, a geometric variational discretization can be …
This work focuses on the derivation and the analysis of a novel, strongly-coupled partitioned method for fluid-structure interaction problems. The flow is assumed to be viscous and incompressible, and the structure is modeled using linear elastodynamics equations. We assume that the structure is thick, i.e., modeled using the same number of spatial dimensions as fluid. Our newly developed numerical method is based on generalized Robin boundary conditions, as well as on the refactorization of the Cauchy's one-legged `theta-like' method, written as a sequence of Backward Euler-Forward Euler step…
In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(τ^2+h^{4-γ}\right)$ with $0<γ<1$, where $τ$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric…
Diffuse Interface Methods for Inverse Problems: Case Study for an Elliptic Cauchy Problem
Institute for Computational and Applied Mathematics, University of Münster（明斯特大学计算与应用数学研究所）；Cells in Motion Cluster of Excellence, University of Münster（明斯特大学细胞运动卓越集群）；Dept. of Mathematical Sciences and Technology, Norwegian University of Life Sciences（挪威生命科学大学数学科学与技术系）
