# A new class of adaptive<br/><br/>discontinuous Petrov-Galerkin (DPG) finite element (FE)<br/><br/>methods with application to singularly perturbed problems

Tuesday, November 2, 2010 - 3:45pm - 4:30pm

Keller 3-180

Leszek Demkowicz (The University of Texas at Austin)

Joint work with Jay Golapalakrishnan, U. Florida.

Adaptive finite elements vary element size h or/and

polynomial order p to deliver approximation properties superior

to standard discretization methods. The best approximation

error

may converge even exponentially fast to zero as a function of

problem size (CPU time, memory). The adaptive methods are thus

a natural

candidate for singularly perturbed problems like

convection-dominated

diffusion, compressible gas dynamics, nearly incompressible

materials, elastic deformation of structures with thin-walled

components, etc. Depending upon the problem, diffusion

constant,

Poisson ratio or beam (plate, shell) thickness, define the

small

parameter.

This is the good news. The bad news is that only a small

number

of variational formulations is stable for adaptive meshes

By the stability we mean a situation where the discretization

error can be bounded by the best approximation error times

a constant that is independent of the mesh. To this class

belong

classical elliptic problems (linear and non-linear),

and a large class of wave propagation problems whose

discretization

is based on hp spaces reproducing the classical exact

grad-curl-div

sequence. Examples include acoustics, Maxwell equations,

elastodynamics,

poroelasticity and various coupled and multiphysics problems.

For singularly perturbed problems, the method should also be

robust, i.e. the stability constant should be independent

of the perturbation parameter. This is also the dream for

wave propagation problems in the frequency domain where the

(inverse of) frequency can be identified as the perturbation

parameter. In this context, robustness implies a method

whose stability properties do not deteriorate with the

frequency

(method free of pollution (phase) error).

We will present a new paradigm for constructing

discretization

schemes for virtually arbitrary systems of linear PDE's that

remain stable for arbitrary hp meshes, extending thus

dramatically

the applicability of hp approximations. The DPG methods build

on two fundamental ideas:

- a Petrov-Galerkin method with optimal test functions for

which

continuous stability automatically implies discrete

stability,

- a discontinuous Petrov-Galerkin formulation based on the

so-called ultra-weak variational hybrid formulation.

We will use linear acoustics and convection-dominated

diffusion

as model problems to present the main concepts and then review

a number of other applications for which we have collected some

numerical experience including:

1D and 2D convection-dominated diffusion (boundary layers)

1D Burgers and compressible Navier-Stokes equations

(shocks)

Timoshenko beam and axisymmetric shells (locking, boundary

layers)

2D linear elasticity (mixed formulation, singularities)

1D and 2D wave propagation (pollution error control)

2D convection and 2D compressible Euler equations (contact

discontinuities and shocks)

The presented methodology incorporates the following features:

The problem of interest is formulated as a system of first

order PDE's in the distributional (weak) form, i.e. all

derivatives

are moved to test functions. We use the DG setting, i.e. the

integration by parts is done over individual elements.

As a consequence, the unknowns include not only field

variables within

elements but also fluxes on interelement boundaries. We do not

use the concept of a numerical flux but, instead, treat the

fluxes as

independent, additional unknowns (a hybrid method).

For each trial function corresponding to either field or

flux

variable, we determine a corresponding optimal test function

by solving an auxiliary local problem on one element.

The use of optimal test functions guarantees attaining the

supremum

in the famous inf-sup condition from Babuska-Brezzi theory.

The resulting stiffness matrix is always hermitian and

positive-definite. In fact, the method can be interpreted as a

least-squares

applied to a preconditioned version of the problem.

By selecting right norms for test functions, we can obtain

stability properties uniform not only with respect to

discretization

parameters but also with respect to the perturbation parameter

(diffusion constant, Reynolds number, beam or shell thickness,

wave number)

In other words, the resulting discretization is robust.

For a detailed presentation on the subject, see [1-8].

[1] L. Demkowicz and J. Gopalakrishnan.

A Class of Discontinuous Petrov-Galerkin Methods.

Part I: The Transport Equation.

Comput. Methods Appl. Mech. Engrg., in print.

see also ICES Report 2009-12.

[2] L. Demkowicz and J. Gopalakrishnan.

A Class of Discontinuous Petrov-Galerkin Methods.

Part II: Optimal Test Functions.

Numer. Mth. Partt. D.E., accepted,

ICES Report 2009-16.

[3] L. Demkowicz, J. Gopalakrishnan and A. Niemi.

A Class of Discontinuous Petrov-Galerkin Methods.

Part III: Adaptivity.

ICES Report 2010-1, submitted to ApNumMath.

[4] A. Niemi, J. Bramwell and L. Demkowicz,

Discontinuous Petrov-Galerkin Method with

Optimal Test Functions for Thin-Body Problems in Solid

Mechanics,

ICES Report 2010-13, submitted to CMAME.

[5] J. Zitelli, I. Muga, L, Demkowicz, J. Gopalakrishnan, D.

Pardo and

V. Calo,

A class of discontinuous Petrov-Galerkin methods. IV:

Wave propagation problems,

ICES Report 2010-17, submitted to J.Comp. Phys.

[6] J. Bramwell, L. Demkowicz and W. Qiu,

Solution of Dual-Mixed Elasticity Equations

Using AFW Element and DPG. A Comparison

ICES Report 2010-23.

[7] J. Chan, L. Demkowicz, R. Moser and N Roberts,

A class of Discontinuous Petrov-Galerkin methods.

Part V: Solution of 1D Burgers and Navier--Stokes

Equations

ICES Report 2010-25.

[8] L Demkowicz and J. Gopalakrishnan, A Class of

Discontinuous

Petrov-Galerkin Methods. Part VI: Convergence Analysis for

the Poisson

Problem, ICES Report, in preparation.

Adaptive finite elements vary element size h or/and

polynomial order p to deliver approximation properties superior

to standard discretization methods. The best approximation

error

may converge even exponentially fast to zero as a function of

problem size (CPU time, memory). The adaptive methods are thus

a natural

candidate for singularly perturbed problems like

convection-dominated

diffusion, compressible gas dynamics, nearly incompressible

materials, elastic deformation of structures with thin-walled

components, etc. Depending upon the problem, diffusion

constant,

Poisson ratio or beam (plate, shell) thickness, define the

small

parameter.

This is the good news. The bad news is that only a small

number

of variational formulations is stable for adaptive meshes

By the stability we mean a situation where the discretization

error can be bounded by the best approximation error times

a constant that is independent of the mesh. To this class

belong

classical elliptic problems (linear and non-linear),

and a large class of wave propagation problems whose

discretization

is based on hp spaces reproducing the classical exact

grad-curl-div

sequence. Examples include acoustics, Maxwell equations,

elastodynamics,

poroelasticity and various coupled and multiphysics problems.

For singularly perturbed problems, the method should also be

robust, i.e. the stability constant should be independent

of the perturbation parameter. This is also the dream for

wave propagation problems in the frequency domain where the

(inverse of) frequency can be identified as the perturbation

parameter. In this context, robustness implies a method

whose stability properties do not deteriorate with the

frequency

(method free of pollution (phase) error).

We will present a new paradigm for constructing

discretization

schemes for virtually arbitrary systems of linear PDE's that

remain stable for arbitrary hp meshes, extending thus

dramatically

the applicability of hp approximations. The DPG methods build

on two fundamental ideas:

- a Petrov-Galerkin method with optimal test functions for

which

continuous stability automatically implies discrete

stability,

- a discontinuous Petrov-Galerkin formulation based on the

so-called ultra-weak variational hybrid formulation.

We will use linear acoustics and convection-dominated

diffusion

as model problems to present the main concepts and then review

a number of other applications for which we have collected some

numerical experience including:

1D and 2D convection-dominated diffusion (boundary layers)

1D Burgers and compressible Navier-Stokes equations

(shocks)

Timoshenko beam and axisymmetric shells (locking, boundary

layers)

2D linear elasticity (mixed formulation, singularities)

1D and 2D wave propagation (pollution error control)

2D convection and 2D compressible Euler equations (contact

discontinuities and shocks)

The presented methodology incorporates the following features:

The problem of interest is formulated as a system of first

order PDE's in the distributional (weak) form, i.e. all

derivatives

are moved to test functions. We use the DG setting, i.e. the

integration by parts is done over individual elements.

As a consequence, the unknowns include not only field

variables within

elements but also fluxes on interelement boundaries. We do not

use the concept of a numerical flux but, instead, treat the

fluxes as

independent, additional unknowns (a hybrid method).

For each trial function corresponding to either field or

flux

variable, we determine a corresponding optimal test function

by solving an auxiliary local problem on one element.

The use of optimal test functions guarantees attaining the

supremum

in the famous inf-sup condition from Babuska-Brezzi theory.

The resulting stiffness matrix is always hermitian and

positive-definite. In fact, the method can be interpreted as a

least-squares

applied to a preconditioned version of the problem.

By selecting right norms for test functions, we can obtain

stability properties uniform not only with respect to

discretization

parameters but also with respect to the perturbation parameter

(diffusion constant, Reynolds number, beam or shell thickness,

wave number)

In other words, the resulting discretization is robust.

For a detailed presentation on the subject, see [1-8].

[1] L. Demkowicz and J. Gopalakrishnan.

A Class of Discontinuous Petrov-Galerkin Methods.

Part I: The Transport Equation.

Comput. Methods Appl. Mech. Engrg., in print.

see also ICES Report 2009-12.

[2] L. Demkowicz and J. Gopalakrishnan.

A Class of Discontinuous Petrov-Galerkin Methods.

Part II: Optimal Test Functions.

Numer. Mth. Partt. D.E., accepted,

ICES Report 2009-16.

[3] L. Demkowicz, J. Gopalakrishnan and A. Niemi.

A Class of Discontinuous Petrov-Galerkin Methods.

Part III: Adaptivity.

ICES Report 2010-1, submitted to ApNumMath.

[4] A. Niemi, J. Bramwell and L. Demkowicz,

Discontinuous Petrov-Galerkin Method with

Optimal Test Functions for Thin-Body Problems in Solid

Mechanics,

ICES Report 2010-13, submitted to CMAME.

[5] J. Zitelli, I. Muga, L, Demkowicz, J. Gopalakrishnan, D.

Pardo and

V. Calo,

A class of discontinuous Petrov-Galerkin methods. IV:

Wave propagation problems,

ICES Report 2010-17, submitted to J.Comp. Phys.

[6] J. Bramwell, L. Demkowicz and W. Qiu,

Solution of Dual-Mixed Elasticity Equations

Using AFW Element and DPG. A Comparison

ICES Report 2010-23.

[7] J. Chan, L. Demkowicz, R. Moser and N Roberts,

A class of Discontinuous Petrov-Galerkin methods.

Part V: Solution of 1D Burgers and Navier--Stokes

Equations

ICES Report 2010-25.

[8] L Demkowicz and J. Gopalakrishnan, A Class of

Discontinuous

Petrov-Galerkin Methods. Part VI: Convergence Analysis for

the Poisson

Problem, ICES Report, in preparation.

MSC Code:

74S05

Keywords: