Abstract – Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDEs as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized.  On requiring the vanishing of the gradient of the Lagrangian with respect to the primal variables, a dual-to-primal mapping from the dual to the primal fields is obtained. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. In this presentation, I will first introduce primal variational forms and then present the essential ingredients of the dual variational formulation as it relates to solving systems of algebraic equations and ODEs/PDEs.  The dual weak form for the linear, one-dimensional, transient convection-diffusion equation will be derived. A Galerkin discretization will be used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either Rectified Power Unit (RePU) activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin method is used with tensor-product B-splines as approximating functions. Numerical results will be presented to demonstrate the accuracy of the method, and rates of convergence will be established in the L2 norm and H1 seminorm for the steady-state convection-diffusion problem.

 

Bio: Sukumar holds a Ph.D. (1998) in Theoretical and Applied Mechanics from Northwestern University. He was a post-doc at Northwestern University and a research associate at Princeton University, before joining UC Davis in 2001, where he is currently a Professor in Civil and Environmental Engineering. Sukumar is a Regional Editor of International Journal of Fracture and a member of the Editorial Boards of Computer Methods in Applied Mechanics and Engineering and Finite Elements in Analysis and Design. Sukumar's current research focuses on maximum-entropy and deep learning based PDE solvers, and developing virtual element methods for the deformation and fracture of solid continua.