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J.S.Howell Research Summary 1 Numerical Analysis and Computation of Fluid/Fluid and Fluid/Structure Interaction Problems Jason S. Howell Fig. 1. Simulation of a coupled ocean-atmosphere system with two-way interaction across a time-averaged interface. In the ocean domain, the stream function is plotted along with the normalized velocity ﬁeld. 1. Overview of Current and Recent Research Activity. In a broad sense, my research interests consist of applying techniques from numerical analysis and computational mathematics to solve problems in applied mathematics. Many of these problems arise from continuum models of physical, geophysical, and biological phenomena, and often result in linear or nonlinear partial diﬀerential equations. These techniques include numerical methods for diﬀerential equations, numerical linear algebra, iterative linear and nonlinear solvers, ﬁnite element and ﬁnite diﬀerence discretization methods, and numerous other topics. I always enjoy learning about and working on new problems as well. Much of my current research is motivated by various aspects of problems arising in the modeling of coupled systems which exhibit dynamic two-way mechanical interaction between the subsystems. Typical examples of real problems that illustrate this behavior include large-scale models of ocean circulation driven in part by interaction with atmospheric motion, and models of blood ﬂow in arteries of large and medium diameter. There is a need for stable and eﬃcient algorithms to estimate solutions to these coupled systems. These problems fall into larger classes of ﬂuid/ﬂuid and ﬂuid/structure interaction problems. At present, few results are available for mathematical and computational modeling of these problems. With these motivating problems in mind, the primary objective of my research is to make meaningful contributions to the analysis and numerical simulation of ﬂuid/ﬂuid and ﬂuid/structure interaction problems. In what follows, I will brieﬂy discuss the main foci of my current research and outline accomplishments made therein. For a more detailed description of my recent, current, and planned research activities, along with a deeper discussion of mathematical background of each area, the interested reader is referred to my web site, http://wwww.math.cmu.edu/∼howell4. 2. Numerical Methods for Fluid/Fluid and Fluid/Structure Interaction. One common ap- proach to the approximation of coupled problems is the use of partitioned methods, which decouple the model equations in each subdomain. In these, each time step involves passing information across the inter- face followed by solving the individual subproblems independently. The subproblems are often approximated using legacy codes, and a major theoretical and computational challenge is the mitigation of timestepping error due to the decoupling. Typical applications in which a partitioned timestepping approach is highly desirable include atmosphere-ocean coupling. • In [5], we investigated a simpliﬁed model of diﬀusion in two adjacent materials which are coupled across a shared and rigid interface through a jump condition. Our investigations produced two ﬁrst- order schemes, a partitioned method and an implicit-explicit method, both of which successfully decouple the subproblems. J.S.Howell Research Summary 2 • In [4], a ﬁrst-order partitioned timestepping method based on a geometric averaging technique for a system consisting of coupled Navier-Stokes ﬂuids with nonlinear interface terms is derived and shown to be unconditionally stable and convergent. The method successfully decouples the global problem into two subdomain problems without a loss of accuracy. • In [3], the partitioned method is extended to the case where multiple time substeps can be used in one subdomain for each single step in the other subdomain, an approach that is well-suited to handle the multiscale dynamics of an ocean-atmosphere system. Blood ﬂow in arteries of medium to large diameter is often modeled as a coupled system with ﬂuid ﬂowing inside a compliant elastic tube. The mechanical interaction between the vessel and the ﬂuid is strongly nonlinear and poses signiﬁcant mathematical and numerical challenges. • In [12], we are analyzing an elastic rod vessel model coupled with a viscoelastic ﬂuid of Johnson- Segalman type as a model of arterial ﬂow. One of the primary objectives of this study is to determine if ﬂuid velocities and stresses in simulations of viscoelastic ﬂows in various arterial geometries diﬀer signiﬁcantly than those obtained when the ﬂuid is assumed to be Newtonian. 3. Finite Element Methods for Fluid and Solid Mechanics. Finite element methods serve as one of the main approaches to the numerical approximation of PDEs arising in models of ﬂuids and elastic structures. Some problems require the use of a mixed method, which introduces an auxiliary (often physically relevant) variable into the variational formulation to enforce a constraint or property, and often results in a saddle point variational problem. An inf-sup condition is necessary for saddle point problems to be well-posed, and in the continuous context, the proof of an inf-sup condition depends on properties of the function spaces and linear operators involved. However, at the discrete level, the inf-sup condition outlines (sometimes restrictive) conditions that approximation spaces must satisfy. Twofold saddle point problems often arise from a dual-mixed variational approach, in which two auxiliary variables are introduced to obtain an alternate formulation of the variational problem. In the case of ﬂuid and elastic problems, the symmetric total stress tensor is one of the auxiliary variables that are introduced. When the constitutive law for the ﬂuid or material is linear (or nonlinear with certain properties), an equivalent problem posed in a pseudostress variable can be posed, relaxing the symmetry requirement of the stress. • In [13], we study well-posedness of twofold saddle point problems derived from PDEs of ﬂuid and solid mechanics. The analysis shows that for these problems, there are multiple sets of equivalent inf-sup conditions, allowing for the use of the most convenient set when showing well-posedness of a particular problem. We also show that twofold saddle point problems can be treated as single saddle point problems, allowing for the immediate application of the existing collection of results for single saddle point problems (such as error estimates) to the twofold case. Additionally, we extend the symmetric tensor Arnold-Winther ﬁnite elements to the nonlinear dual-mixed Stokes problem and prove a new inf-sup condition that divergence-free Arnold-Winther tensors satisfy. • In [10], I show that the dual-mixed Stokes problem can be approximated by using the pseudostress and trace-free velocity gradients. • In [8], I extend this approach to the generalized Stokes problem (with a zeroth-order velocity term) and reduce by one third the overall degrees of freedom needed by an existing method. New penalty methods for dual-mixed Stokes problems are also derived there. • In [1], we are hybridizing the dual-mixed methods in [8]. The only globally coupled degrees of freedom in the resulting hybrid method are those of an approximation to the velocity on the boundaries of the individual elements in the triangulation, and the linear system that results is symmetric positive deﬁnite. 4. Numerical Methods for Non-Newtonian Fluids. Non-Newtonian ﬂuids exhibit a complicated stress-strain relationship that may contain nonlinearities or material derivatives of the extra stress. Two types of non-Newtonian ﬂuids often found in industrial and biological applications are shear-thinning ﬂuids and viscoelastic ﬂuids. Approximating solutions to mathematical models of non-Newtonian ﬂuid ﬂow requires extensive computational eﬀort. This can be due to (a) the modeling systems consisting of multiple unknowns (such as velocity, pressure, and stress), (b) nonlinearities in the system needing to be handled appropriately, (c) the ﬂow or the ﬂow domain often having characteristics that require a ﬁne computational mesh in one or more locations, or many other aspects of the problem. Shear-thinning ﬂuids exhibit a decreasing viscosity J.S.Howell Research Summary 3 with increased shear rate. • In [7], we analyze a dual-mixed formulation for a general class of shear-thinning ﬂuids under steady- state and creeping (inertialess) ﬂow assumptions. We show that the dual-mixed method, posed in the appropriate function spaces, is well-posed in the continuous and discrete cases, and derive error estimates that are dependent on problem parameters. The dual-mixed approach for nonlinear Stokes ﬂows is well-suited for adaptive computation, as the regularity requirements for approximation spaces are relaxed. • In [11], I am investigating residual a posteriori error estimators for the method in [7], based on the technique of nonlinear projections of residual functions. The actual nonlinear projections themselves do not need to be computed; instead, the error estimates are given in terms of the norms of the residual functions. Viscoelastic ﬂuids are characterized by the ability of the ﬂuid particles to retain and release elastic energy. A large class of viscoelastic ﬂuids can be described by the Johnson-Segalman constitutive law, which contain a parameter (the Weissenberg number) that describes in part the elastic behavior of ﬂuid particles. One issue arising in the approximation of these ﬂuids is the High Weissenberg Number Problem, which manifests itself numerically as nonlinear solvers fail to converge for Weissenberg numbers larger than some critical value. • In [6], we have developed a two-parameter defect correction method for Johnson-Segalman viscoelas- tic ﬂuid ﬂows. We show well-posedness of the method along with error estimates and, for some sets of problem parameters, the defect-correction method can extend the range of Weissenberg number for which solutions to the problem can be computed. • In [9], I describe the implementation of pseudo-arclength continuation methods, using the Weis- senberg number as a continuation parameter, for the steady Johnson-Segalman problem and use the methods to compute approximate solutions for increasing values of the Weissenberg number. I have also recently begun work with N. Walkington on a reformulation of the modeling equations of a viscoelastic ﬂuid into a structure that better preserves the kinematic behavior of the ﬂuid. One of the objectives of this approach is to describe the ﬂuid constitutive law in a manner that avoids the numerical diﬃculties encountered in approximating high Weissenberg number ﬂows. 5. Short and Long Term Objectives. Many of my current research activities have been illustrated above. In the near future, I plan to consider further extensions and applications of this work such as: • Construction and theoretical justiﬁcation of higher-order decoupling schemes for the two-domain parabolic problem and the coupled ocean-atmosphere model. • In [2], we are performing computational studies of diﬀerent ocean-atmosphere geometries using the partitioned method of [4]. • Analysis and computations of arterial blood ﬂow using shear-thinning constitutive models, and extensions to 3-D ﬂuid/3-D structure computations. • Numerical studies involving the Arnold-Winther tensors for the Stokes problem. • Condition estimates and preconditioning strategies for the dual-mixed methods for the steady and generalized Stokes problems. • Dual-mixed approaches for elasticity and ﬂuids with new ﬁnite elements designed for weak symmetry. • Dual-mixed approaches for the vanishing moment method for the Monge-Amp`re equation. e In the long term, I seek to merge many aspects of my current activities into methods that can be applied to challenging problems. One example of this is to extend the dual-mixed approach to the nonlinear Navier- Stokes equations, viscoelastic ﬂuids, time-dependent ﬂuid and elasticity problems, and coupled models. In closing, I reiterate that I am always interested in new problems and I thoroughly enjoy collaborating with scientists from all ﬁelds. REFERENCES [1] B. Cockburn and J. S. Howell, Hybridization of a dual mixed method for Stokes problems, in preparation (2009). [2] J. M. Connors and J. S. Howell, Numerical studies of coupled ocean-atmosphere models with two-way interaction, in preparation (2009). [3] J. M. Connors, J. S. Howell, and W. J. Layton, Decoupled time stepping for ﬂuid-ﬂuid interaction with variable subproblem step sizes, in preparation (2009). [4] , Decoupled time stepping methods for ﬂuid-ﬂuid interaction, SIAM J. Numer. Anal., submitted (2009). J.S.Howell Research Summary 4 [5] , Partitioned timestepping for a two-domain parabolic problem, SIAM J. Numer. Anal., to appear (2009). [6] V. J. Ervin, J. S. Howell, and H. Lee, A two-parameter defect-correction method for computation of steady-state viscoelastic ﬂuid ﬂow, Appl. Math. Comput., 196 (2008), pp. 818–834. [7] V. J. Ervin, J. S. Howell, and I. Stanculescu, A dual-mixed approximation method for a three-ﬁeld model of a nonlinear generalized Stokes problem, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 2886–2900. [8] J. S. Howell, Approximation of generalized and steady Stokes problems using dual-mixed ﬁnite elements without enrich- ment, Int. J. Numer. Meth. Fluids, submitted (2009). [9] J. S. Howell, Computation of viscoelastic ﬂuid ﬂows using continuation methods, J. Comput. Appl. Math., 225 (2009), pp. 187–201. [10] J. S. Howell, Dual-mixed ﬁnite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients, J. Comput. Appl. Math., 231 (2009), pp. 780–792. [11] , A posteriori error estimates and adaptive computation of a dual-mixed approximation method for a nonlinear Stokes problem, in preparation (2009). [12] J. S. Howell and H. Lee, Numerical approximation of viscoelastic ﬂows in an elastic medium, in preparation (2009). [13] J. S. Howell and N. J. Walkington, Inf-sup conditions for twofold saddle point problems, Numer. Math., submitted (2009).