Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions. Space-Time Formulations and Elements of Irregular Shapes

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The project focuses on the development of a new technique for simulating complicated physical and engineering processes with computers - the Discontinuous Petrov Galerkin (DPG) method. The DPG methodology has the potential to transform existing Finite Element software at US National Labs and industry. Without undermining the logic of existing codes, the method asks for a complete overhaul of local operations (on the element level) utilizing emerging multicore architectures. The method will have a lasting impact on software development for difficult problems arising in applications which require high accuracy simulations. Application areas targeted in this work include aerospace engineering (transonic and supersonic flows), geomechanics and bioengineering (implants, imaging of soft tissues). On the application side, the project builds on collaboration with Sandia and Argonne National Laboratories, Ben Gurion University and Boeing.The project focuses on two research directions: (a) DPG space time elements, and (b) use of elements with irregular shapes. The DPG method minimizes residuals in the dual norm corresponding to a prespecified test norm. Computation of the residual requires inversion of the Riesz operator in the test space. With the use of broken test spaces and localizable test norms, the inversion can be done element-wise using standard Galerkin and "enriched" spaces. With the error of inverting the Riesz operator controlled locally, i.e. on the element level, the method automatically guarantees discrete stability for any well-posed problem in the sense of classic theory of closed operators. The methodology leads to uniform stability for singular perturbation problems and, being a minimization method, does not suffer from any preasymptotic instabilities. The residual is computed rather than estimated and provides a basis for automatic adaptivity. The first focus area deals with space-time elements for convection-dominated problems with applications to compressible and incompressible Navier-Stokes (NS) equations in two space dimensions. The second line of research investigates the sensitivity of the method to element shapes with a view to using elements of arbitrary polyhedral shapes. This part of the project will be done in the context of linear elasticity and hyperelasticity in three space dimensions with applications to geomechanics and bioengineering. The proposed work includes both analysis and semi-professional software development. The project builds on existing codes and joint computational effort with Argonne and Sandia National Laboratories.