Quasi-optimal and pressure robust discretizations of the Stokes equations by new augmented Lagrangian Formulations See

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure robust, in the sense that the velocity H^1-error is proportional to the best H^1-error to the analytical velocity. This shows that such a property can be achieved without using conforming and divergence-free pairs. We bound also the pressure ...

Convergence of adaptive C0-interior penalty Galerkin method for the biharmonic problem See

We develop a basic convergence analysis for an adaptive C0IPG method for the Biharmonic problem which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoul...

Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations See

Multigrid methods belong to the best-known methods for solving linear systems arising from the discretization of elliptic partial differential equations. The main attraction of multigrid methods is that they have an asymptotically meshindependent convergence behavior. Multigrid with Vanka (or local multilevel pressure Schur complement method) as smoother have been frequently used for the construction of very effcient coupled monolithic solvers for the solution of the stationary incompre...

A quasi-optimal Crouzeix-Raviart discretization of the Stokes equations See

Algebraic limiting techniques and hp-adaptivity for continuous finite element discretizations See

Bulk-robust assignment problems: hardness, approximability and algorithms See

This thesis studies robust assignment problems with focus on computational complexity. Assignment problems are well-studied combinatorial optimization problems with numerous practical applications, for instance in production planning. Classical approaches to optimization expect the input data for a problem to be given precisely. In contrast, real-life optimization problems are modeled using forecasts resulting in uncertain problem parameters. This fact can be taken into account using the f...

A partition of unity approach to adaptivity and limiting in continuous finite element methods See

The partition of unity finite element method (PUFEM) proposed in this paper makes it possible to blend space and time approximations of different orders in a continuous manner. The lack of abrupt changes in the local mesh size h and polynomial degree p simplifies implementation and eliminates the need for using sophisticated hierarchical data structures. In contrast to traditional hp-adaptivity for finite elements, the proposed approach preserves discrete conservation properties and the conti...

Gradient-based limiting and stabilization of continuous Galerkin methods See

In this paper, we stabilize and limit continuous Galerkin discretizations of a linear transport equation using an algebraic approach to derivation of artificial diffusion operators. Building on recent advances in the analysis and design of edge-based algebraic flux correction schemes for singularly perturbed convection-diffusion problems, we derive algebraic stabilization operators that generate nonlinear high-order stabilization in smooth regions and enforce discrete maximum principles every...

Sensitivity analysis of elliptic variational inequalities of the first and the second kind See

This thesis is concerned with the differential sensitivity analysis of elliptic variational inequalities of the first and the second kind in finite and infinite dimensions. We develop a general theory that provides a sharp criterion for the Hadamard directional differentiability of the solution operator to an elliptic variational inequality and introduce several tools that facilitate the sensitivity analysis in practical applications. Our analysis is accompanied by examples from mechanics ...

Frame-invariant directional vector limiters for discontinuous Galerkin methods See

Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertexbased slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector fi...

A flux-corrected RBF-FD method for convection dominated problems in domains and on manifolds See

In this article we introduce a FCT stabilized Radial Basis Function (RBF)-Finite Difference (FD) method for the numerical solution of convection dominated problems. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the solution for an almost random placement of scattered data nodes. The method can be applicable both for problems defined in a domain or if equipped with level set techniques, on a stationary manifold. We demonstrate the numerical beh...

Bathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization See

In the present paper, we use modified shallow water equations (SWE) to reconstruct the bottom topography (also called bathymetry) of a flow domain without resorting to traditional inverse modeling techniques such as adjoint methods. The discretization in space is performed using a piecewise linear discontinuous Galerkin (DG) approximation of the free surface elevation and (linear) continuous finite elements for the bathymetry. Our approach guarantees compatibility of the discrete forward and ...

A monolithic conservative level set method with built-in redistancing See

We introduce a new level set method for representing evolving interfaces. In the case of divergence-free velocity fields, the new method satisfies a conservation principle. Conservation is important for many applications such as modeling two-phase incompressible flow. In the present implementation, the conserved quantity is defined as the integral of a smoothed characteristic function. The new approach embeds level sets into a volume of fluid formulation. The evolution of an approximate signe...

Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields See

This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to ...

Convergence of adaptive finite element methods with error-dominated oscillation See

Recently, we devised an approach to a posteriori error analysis, which clarifies the role of oscillation and where oscillation is bounded in terms of the current approximation error. Basing upon this approach, we derive plain convergence of adaptive linear finite elements approximating the Poisson problem. The result covers arbritray H^-1-data and characterizes convergent marking strategies.

An Investigation on Separation Points of Power-Law model Along a Rotating Round-Nosed Body See

The purpose of present study is to numerically investigate the natural convection flow of Ostwalde-de Waele type power law non-Newtonian fluid along the surface of rotating axi-symmetric round-nosed body. For computational purpose rotating hemisphere is used as a case study in order to examine the heat transfer mechanism near such transverse curvature geometries. The numerical scheme is applied after converting the dimensionless system of equations into primitive variable formulations. Implic...

On the Prospects of Using Machine Learning for the Numerical Simulation of PDEs: Training Neural Networks to Assemble Approximate Inverses See

In an unconventional approach to combining the very successful Finite Element Methods (FEM) for PDE-based simulation with techniques evolved from the domain of Machine Learning (ML) we employ approximate inverses of the system matrices generated by neural networks in the linear solver. We demonstrate the success of this solver technique on the basis of the Poisson equation which can be seen as a fundamental PDE for many practically relevant simulations [Turek 1999]. We use a basic Richardson ...

Numerical Benchmarking for 3D Multiphase Flow: New Results for a Rising Bubble See

Based on the benchmark results in [1] for a 2D rising bubble, we present the extension towards 3D providing test cases with corresponding reference results, following the suggestions in [2]. Additionally, we include also an axisymmetric configuration which allows 2.5D simulations and which provides further possibilities for validation and evaluation of numerical multiphase flow components and software tools in 3D.

Numerical study of the RBF-FD level set based method for partial differential equations on evolving-in-time surfaces See

In this article we present a Radial Basis Function (RBF)-Finite Difference (FD) level set based method for numerical solution of partial differential equations (PDEs) of the reaction-diffusion-convection type on an evolving-in-time hypersurface Γ (t). In a series of numerical experiments we study the accuracy and robustness of the proposed scheme and demonstrate that the method is applicable to practical models.