High order numerical methods for multiphysics couplings

该项目的结果在复杂的流动和运输问题的高阶间断伽辽金方法的发展。研究人员分析和数值研究不同类型的耦合，如斯托克斯或Navier-Stokes方程与混相位移和多相流的耦合。在表面区域和次表面区域之间的界面处施加物理上有意义的条件。数值分析中的数学进步包括证明耦合系统高阶解的存在性和唯一性，以及严格推导关于网格大小和多项式次数的先验误差估计。 数值解的验证是通过收敛性研究，相对于网格大小和多项式的程度和比较与其他离散技术得到的解决方案的自适应性。开发了适用于二维和三维问题的计算框架，这项工作涉及多物理场问题的建模、分析和模拟，例如河流和湖泊的地下水污染。当人造物质或有时是自然产生的物质溶解在补给地下水的沃茨中时，地下水就会受到污染。 由于地下水与湖泊和河流相连，这些地表沃茨的污染往往意味着含水层的污染。研究人员研究地表流和地下流之间的相互作用。 精确和有效的数值方法的开发和分析。密集的数值模拟和与实验数据的比较，可以获得更多的知识多物理场耦合。 该项目的跨学科性质促进了工程和科学界之间的科学知识转移。该项目通过培训几名研究生和本科生，在促进学习的同时，促进了发现和理解。 一项重要的教育活动是为进入10-12年级的高中生提供为期一周的夏季数学学校。活动？的目标是鼓励学生追求科学和数学的职业生涯。

The project results in the development of high order discontinuous Galerkin methods for complex flow and transport problems. The investigator analyzes and numerically studies different types of couplings, such as the coupling of Stokes or Navier-Stokes equations with miscible displacement and multiphase flow. Physically meaningful conditions are imposed at the interface between surface region and subsurface region. The mathematical advances in numerical analysis consist of proving existence and uniqueness of high order solutions of coupled systems and of rigorously deriving a priori error estimates with respect to the mesh size and the polynomial degree. Verification of the numerical solutions is performed through convergence studies, adaptivity with respect to the mesh size and the polynomial degree and comparisons with solutions obtained from other discretization techniques. A computational framework valid for two and three-dimensional problems is developed.This work deals with the modeling, analysis and simulation of multiphysics problems arising for instance in groundwater contamination through rivers and lakes. Groundwater becomes contaminated when human-made substances or sometimes naturally occurring substances are dissolved in waters recharging the groundwater. Often, as groundwater is connected with lakes and rivers, the pollution of these surface waters implies the pollution of aquifers. The investigator studies the interaction between surface flow and subsurface flow. Accurate and efficient numerical methods are developed and analyzed. Intensive numerical simulations and comparison with experimental data allow gaining further knowledge on multiphysics couplings. The interdisciplinary nature of the task fosters transfer of scientific knowledge across the engineering and science communities.This project advances discovery and understanding while promoting learning by training several graduate students and undergraduate students. An important educational activity is a week-long Summer Math School offered to high-school students entering grades 10-12. The event?s objective is to encourage students to pursue a career in science and mathematics.