John A. Trangenstein

Duke University

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John B. Bell

Lawrence Berkeley National Laboratory

Phillip Colella

Lawrence Berkeley National Laboratory

Michael Welcome

Lawrence Berkeley National Laboratory

Richard B. Pember

Lawrence Livermore National Laboratory

Richard D. Hornung

Lawrence Livermore National Laboratory

Gregory R. Shubin

Boeing (United States)

F.Xabier Garaizar

Lawrence Livermore National Laboratory

Cass T. Miller

University of North Carolina at Chapel Hill

Joseph A. Pedit

University of North Carolina at Chapel Hill

Malcolm Reed

Brighton and Sussex Medical School

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Lawrence Livermore National Laboratory

University of North Carolina at Chapel Hill

Duke University

University of California, Berkeley

Lawrence Berkeley National Laboratory

Naval Surface Warfare Center

The University of Texas at Austin

Princeton University

University of California, Davis

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Mixed and Hybrid Finite Elements

Cambridge University Press eBooks (2013)

John A. Trangenstein

Our discussion of conforming finite element methods in Chapters 4, 5 and 6 showed how to develop arbitrarily high-order numerical approximations for smooth problems. As we saw in Section 6.4.1, these numerical approximations were required to be continuous enough to be admissible in the weak form for the differential equation. In Céa's Lemma 5.5.1, we also saw that these conforming finite solutions consistently overestimate the strain energy in the true solution of the differential equation. This is an unfortunate consequence of the development of conforming finite element methods from energy minimization principles.

Lemma (botany)

Section (typography)

Strain energy

Minification

10.1017/cbo9781139025508.008

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Citations (115)

Introduction to Partial Differential Equations

Cambridge University Press eBooks (2013)

John A. Trangenstein

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Content (measure theory)

10.1017/cbo9781139025508.002

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Citations (79)

Finite Element Theory

Cambridge University Press eBooks (2013)

John A. Trangenstein

Our goal in this chapter is to analyze the well-posedness of finite element methods. Of course, the sensitivity of numerical methods to perturbations in their data is governed partly by the sensitivity of the underlying equations being solved. Thus we will need to study the existence, uniqueness and continuous dependence on the data for elliptic boundary value problems. These issues will be described in terms of some basic tools of functional analysis, beginning with function norms and spaces in Sections 5.1.1 and 5.1.2. Appropriate function spaces will allow us to describe function derivatives in Section 5.1.3, and function norms that involve derivatives in Section 5.2. These derivative norms allow us to bound the solution of elliptic partial differential equations in terms of features of the problem domain, coefficients in the differential equation and boundary values. Once we have described elliptic equations in Section 5.3 and developed some theory of well-posedness for elliptic boundary value problems in Section 5.4, we will be able to discuss some basic ideas regarding the convergence of finite element methods in Section 5.5. However, some important details of this finite element convergence theory will be delayed until Chapter 6. Those details concern polynomial approximation, coordinate maps, domain approximation, quadrature rules, and numerical treatment of boundary conditions.

Section (typography)

10.1017/cbo9781139025508.006

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Parabolic Equations

Cambridge University Press eBooks (2013)

John A. Trangenstein

In this chapter we will study finite difference methods for the numerical solution of parabolic partial differential equations. Finite difference methods are simple to describe for problems on rectangular domains, but are difficult to extend to general domains or to high order. We will see that explicit time integration of these methods will lead to severe restrictions on the timestep for numerical stability, but implicit treatment will require the solution of large sparse systems of equations. We will also see that finite differences do not easily develop second-order discretizations of Neumann boundary conditions. Our discussions of implicit methods in this chapter will motivate later development of iterative methods for solving linear systems of equations in Chapter 3.

Finite difference

10.1017/cbo9781139025508.003

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Higher order Godunov methods for general systems of hyperbolic conservation laws

Journal of Computational Physics (1989)

John B. Bell Phillip Colella John A. Trangenstein

Godunov's scheme

Conservation law

Roe solver

10.1016/0021-9991(89)90054-5

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Citations (176)

Front Tracking for Shear Bands in an Antiplane Shear Model

Journal of Computational Physics (1997)

F.Xabier Garaizar John A. Trangenstein

Shear band

10.1006/jcph.1996.5456

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Citations (4)

Scientific Computing: Vol. III - Approximation and Integration

John A. Trangenstein

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Interpolation and Approximation

Texts in computational science and engineering (2017)

John A. Trangenstein

Trigonometric interpolation

Interpolation

Birkhoff interpolation

Hermite interpolation

Trigonometric polynomial

Lagrange polynomial

Box spline

Chebyshev nodes

10.1007/978-3-319-69110-7_1

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Citations (69)

Finite Element Approximations

Cambridge University Press eBooks (2013)

John A. Trangenstein

In Chapter 5 we used some important results from functional analysis to prove some convergence theorems for finite element methods. Careful readers of that chapter will notice at least two very important unresolved issues. First, we did not prove that the approximation assumption in (5.58) is satisfied for any finite dimensional set of functions. To resolve this issue in this chapter, we will discuss the approximation assumption in Section 6.3. Second, in finite element methods, we approximate the weak solution of partial differential equations by mapped shape functions. For unit shapes other than triangles or tetrahedra and for nonlinear coordinate mappings, the mapped shape functions are generally not polynomials, even though the shape functions are polynomials before applying the coordinate maps. We will analyze the error in finite element approximations employing coordinate transformations in Sections 6.4.2 and 6.4.3.

Tetrahedron

Section (typography)

Approximations of π

10.1017/cbo9781139025508.007

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Citations (3)

Scientific Computing: Vol. II - Eigenvalues and Optimization

John A. Trangenstein

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