After motivating the introduction of the so-called hybridizable discontinuous Galerkin (HDG) methods, we give an overview of their development since their inception. This will be done in the framework of steady-state diffusion problems. Extensions to other problems will be also reviewed.
We present an overview on analysis and implementation
of high order, in particular hp, Finite Element Methods (FEM),
with particular attention to
a) singular solutions,
b) analytic regularity theory, and
c) exponential convergence rate bounds of DG and CG versions of hp-FEM in three dimensions.
d) Recent results and ongoing research on exponential convergence of
quantized tensor-train (QTT)
formatted first order FEM based on a) , b) and c) will also be given.
a,b,c) are joint work with Dominik Schoetzau (Vancouver) and Thomas Wihler (Bern), d) is joint work with Vladimir Kazeev (Geneva)
We introduce a family of preconditioners for linear systems of equations arising from a symmetric Interior Penalty Discontinuous Galerkin (IP-DG) discretization of Hcurl(Ω)-elliptic boundary value problems. The design and analysis of the preconditioners relies on the auxiliary space method (ASM) employing an auxiliary space of Hcurl(Ω)-conforming finite element functions together with a relaxation technique. We analyze the asymptotic convergence of the proposed preconditioners addressing specifically the robustness of the solvers with respect to jumps in the coefficients of the second and zeroth order parts of the operator, respectively. Extensive numerical experiments are included to verify the theory and asses the performance of the preconditioners.
Joint work with RalfHiptmair and Cecilia Pagliantini from ETH, Zürich.
Simulation of flow and transport processes in porous media provides a formidable challenge and application field for high-performance computing. Relevant continuum-scale models include partial differential equations of elliptic, parabolic and hyperbolic type which are coupled through highly nonlinear coefficient functions. The multi-scale character and uncertainties in the parameters constitute an additional level of complexity but provide also opportunities for high-performance computing.
This talk will focus on the efficient solution of porous media flow problems with discontinuous Galerkin methods. For the fast solution of the linear systems arising in the elliptic flow equation a hybrid preconditioner based on subspace correction in the conforming finite element subspace is employed. The transport equation is solved with explicit methods as part of an operator splitting scheme for the coupled system. The performance of the implementation based on the sum factorization approach is assessed and computational results for unstable flows in porous media are presented.
Depending on how each equation in the Maxwell system is treated (weakly or strongly), one gets a proliferation of variational formulations for a single electric cavity boundary value problem. We will show that stability of one variational formulation implies the stability of all others. Then we discuss the stability of their DPG discretizations.
My collaboration with Dominik started over ten years ago, when we considered preonditioners for the saddle-point formulation of the time-harmonic Maxwell equations. The mathematical problem is fascinating, and the discrete Helmholtz equation does magic that allowed us to develop an effective block preconditioner, and some matrix analysis to go along with it. It's been fun to work together ever since. Recently, my student Ron Estrin and I have taken a broader look at the algebraic properties of the class of nonsingular saddle-point matrices whose leading block is maximally rank deficient, and have found that the inverse in this case has unique mathematical properties. This allows for developing a class of indefinite block preconditioners that rely on approximating the null space of the leading block. The preconditioned matrix is a product of two indefinite matrices, but under certain conditions the conjugate gradient method can be applied and is rapidly convergent. Spectral properties of the preconditioners are observed and validated by numerical experiments.
The accurate and reliable simulation of wave phenomena is of fundamental importance in a wide range of engineering and medical applications such as seismic tomography, wireless communication, ultrasound imaging, and non-invasive testing. To address the wide range of difficulties involved, we consider symmetric interior penalty discontinuous Galerkin (IP-DG) methods for the wave equation (Grote, Schneebeli, Schötzau, 2006, 2009), which easily handle elements of various types and shapes, irregular non-matching grids, and even locally varying polynomial order. Moreover, in contrast to standard (conforming) finite element methods, IP-DG methods yield an essentially diagonal mass matrix; hence, when coupled with explicit time integration, the overall numerical schemes remain truly explicit in time.
Local mesh refinement severely impedes the efficiency of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods overcome the bottleneck due to a few small elements by allowing smaller time-steps precisely where those elements are located. Yet when the region of local mesh refinement itself contains a sub-region of even smaller elements, any local time-step again will be overly restricted. To remedy the repeated bottleneck caused by hierarchical mesh refinement, multilevel local time-stepping methods are proposed, which permit the use of the appropriate time-step at every level of mesh refinement. Based on the LTS methods from Diaz and Grote (2009), these multi-level LTS methods are explicit, yield arbitrarily high accuracy and conserve the energy.
Motivated by boundary value problems from magneto-hydrodynamics we study advection problems for differential forms. They are stated in terms of the Lie derivative in a given velocity field. They also represent a family of transport problems, of which scalar advection is one member.
Inspired by successful schemes devised for the scalar case we pursue a stabilized Galerkin approach in the spirit of discontinuous Galerkin methods with upwind numerical flux. We note that even if the solution is approximated by means of discrete differential forms, jump terms across interelement faces have to be retained for non-scalar problems, and they hold the key to stability. Rigorous a priori convergence estimate are provided for the stationary problem and for Lipschitz continuous velocity fields.
For discontinuous velocities, existence of solutions of is open beyond the scalar case. However, an extension of a stabilized Galerkin scheme performs well in numerical experiments also in this case.
In this talk we consider the hp-version interior penalty discontinuous Galerkin method (DGFEM) for the discretization of second order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements. By admitting such general meshes, this class of methods allows for the approximation of problems posed on com- putational domains which may contain a huge number of local geometrical features, or micro-structures. While standard numerical methods can be de- vised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying (composite) finite element space based on general polytopic meshes is inde- pendent of the number of geometric features. Here, we consider both the a priori and a posteriori error analysis of this class of methods, as well as their application within Schwarz-type domain decomposition preconditioners.
This is joint work with Paola Antonietti (MOX, Milan), Andrea Cangian (Leicester), Joe Collis (Nottingham), Zhaonan Dong (Leicester), Manolis Georgoulis (Leicester) and Stefano Giani (Durham).
We consider the approximation of solutions of singularly perturbed elliptic equations. Two classes are studied. The first class consists of elliptic-elliptic problems. There, the classical Galerkin approximation is optimal in the so-called energy norm, which is, however, so weak that features within the layer are not well resolved. We show that robust exponential convergence of a high order FEM on suitable meshes is achieved also in a stronger, ``balanced'' norm. This latter norm is balanced in the sense that layer contributions are of size $O(1)$ uniformly in the perturbation parameter. The second class consists of systems of singularly perturbed ODEs of elliptic-hyperbolic type with multiple scales. We discuss well-posedness of the variational formulations and robust convergence of an $h$-version Discontinuous Galerkin method on Shishkin meshes.
We introduce a generalization of the artificial compressibility method for approximation of the incompressible Navier-Stokes equations. It allows for the construction of schemes of any order in time that require the solution of a fixed number of vectorial parabolic problems, depending only on the desired order of the scheme. These problems have a condition number that scales like k h-2, with k being the time step and h being the spatial grid size. This approach has several advantages in comparison to the traditional pro- jection schemes widely used for the unsteady Navier-Stokes equations. First, it allows for the construction of schemes of any order for both, the veloc- ity and pressure, while the best proven accuracy achievable by a projection scheme is second order on the velocity and 3/2 order on the pressure. Second, the projection schemes require the solution of an elliptic scalar problem for the pressure that has a condition number of order h-2 in addition to a vectorial parabolic problem for the velocity. This makes them slower if iterative meth- ods are used to solve the linear systems. Third, the artificial compressibility schemes of second or higher order based on a defect correction approach pre- sented below have superior stability properties as compared to a second order projection scheme. Finally, we will discuss the approach for solving the linear system resulting from the momentum equation and more particularly, we will present an unconditionally stable scheme that treats the mixed derivatives in the grad div operator fully explicitly.
Following both theoretical analysis and extensive numerical testing, Trefftz Discontinuous Galerkin (TDG) methods have shown promise for efficiently solving time harmonic wave equations including the Helmholtz equation, Maxwell?s equations and Liner Elasticity. However, one drawback is that coefficients describing the physical medium (for example, refractive index) need to be piecewise constant. In some applications spatially dependent coefficients are encountered. In this talk I shall describe a technique for generating approximate local solutions of the Helmholtz equation with non-constant coefficients called Generalized Plane Waves. To use these functions with TDG requires a generalization of the standard TDG scheme. An error analysis will be described and numerical examples will be presented.
The response of the muscle-tissue unit (MTU) to activation and applied forces is affected by the architectural details as well as the material properties of this nearly-incompressible tissue. We will describe the (highly nonlinear) elastic equations governing this response for a fully three-dimensional, quasi-static, fully nonlinear and anisotropic MTU. We describe a three-field formulation for this problem, and present a DG discretization strategy. The scheme was implemented using {\tt deal.ii}. We present computational results about the effects of localized activation as well as the effects of fatty tissue on muscle response. This is joint with Sebastian Dominguez, Hadi Rahemi, David Ryan and James Wakeling.
In this talk we present a new mixed-type finite element method for the Navier-Stokes problem. This new method is based on the introduction of a modified pseudostress tensor depending nonlinearly on the velocity through the respective convective term. The pressure is eliminated by using the incompressibility condition. Next, the equations are augmented with Galerkin type terms arising from the constitutive and equilibrium equations, and from the Dirichlet boundary condition. In this way, we propose an augmented mixed method for the fluid flow problem where the only unknowns are given by the aforementioned modified pseudostress and the velocity. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Then, the classical Banach’s fixed point Theorem and Lax-Milgram’s Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea’s estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k + 1 for the velocity, which leads to an optimal convergent scheme. Finally, we extend our analysis to the Navier-Stokes problem with variable viscosity and the Boussinesq equations.
A space-time discontinuous Galerkin method for wave propagation problems will be presented and analysed. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space-time) mesh.
For acoustic and electromagnetic wave equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces, and high order error bounds in the meshwidth and in the polynomial degree. The analysis framework also applies to the case of higher space dimensions. Some numerical experiments demonstrate the theoretical results and the faster convergence compared to the non-Trefftz version of the scheme.
These results have been obtained in collaboration with Fritz Kretzschmar (TU Darmstadt), Andrea Moiola (University of Reading), and Sascha M. Schnepp (ETH Zurich).
In this talk I will present some preliminary results on the use of an HDG method for the simulation of elastic waves. I will show how the Qiu and Shi choice of spaces and stabilization parameters for an HDG scheme applied to quasi-static elasticity also apply for time harmonic elastic waves, providing a superconvergent method. I will next discuss a conservation of energy property that holds in the transient case when the elasticity equations are semidiscretized in space with the same HDG strategy.
This work is a collaboration with Allan Hungria (University of Delaware) and Daniele Prada (Indiana University Purdue University at Indianapolis)
We consider stabilized mixed finite element methods for the Stokes problem. Previous error analysis of these methods have assumed that the solution is smooth. Using a technique of T. Gudi we now derive a quasioptimal error estimate, i.e. we show that the error is bounded by the interpolation error and an data oscillation term. In addition, we derive an residual based a posteriori error.
We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. We develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop an adaptive time step algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.