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| List of Figures | |
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| List of Tables | |
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| Preface | |
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| Acknowledgments | |
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| Partial Differential Equations | |
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| Selected general properties | |
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| Classification and examples | |
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| Hadamard's well-posedness | |
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| General existence and uniqueness results | |
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| Exercises | |
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| Second-order elliptic problems | |
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| Weak formulation of a model problem | |
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| Bilinear forms, energy norm, and energetic inner product | |
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| The Lax-Milgram lemma | |
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| Unique solvability of the model problem | |
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| Nonhomogeneous Dirichlet boundary conditions | |
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| Neumann boundary conditions | |
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| Newton (Robin) boundary conditions | |
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| Combining essential and natural boundary conditions | |
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| Energy of elliptic problems | |
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| Maximum principles and well-posedness | |
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| Exercises | |
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| Second-order parabolic problems | |
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| Initial and boundary conditions | |
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| Weak formulation | |
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| Existence and uniqueness of solution | |
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| Exercises | |
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| Second-order hyperbolic problems | |
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| Initial and boundary conditions | |
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| Weak formulation and unique solvability | |
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| The wave equation | |
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| Exercises | |
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| First-order hyperbolic problems | |
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| Conservation laws | |
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| Characteristics | |
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| Exact solution to linear first-order systems | |
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| Riemann problem | |
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| Nonlinear flux and shock formation | |
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| Exercises | |
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| Continuous Elements for 1D Problems | |
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| The general framework | |
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| The Galerkin method | |
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| Orthogonality of error and Cea's lemma | |
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| Convergence of the Galerkin method | |
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| Ritz method for symmetric problems | |
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| Exercises | |
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| Lowest-order elements | |
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| Model problem | |
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| Finite-dimensional subspace V[subscript n subset or is implied by] V | |
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| Piecewise-affine basis functions | |
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| The system of linear algebraic equations | |
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| Element-by-element assembling procedure | |
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| Refinement and convergence | |
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| Exercises | |
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| Higher-order numerical quadrature | |
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| Gaussian quadrature rules | |
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| Selected quadrature constants | |
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| Adaptive quadrature | |
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| Exercises | |
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| Higher-order elements | |
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| Motivation problem | |
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| Affine concept: reference domain and reference maps | |
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| Transformation of weak forms to the reference domain | |
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| Higher-order Lagrange nodal shape functions | |
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| Chebyshev and Gauss-Lobatto nodal points | |
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| Higher-order Lobatto hierarchic shape functions | |
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| Constructing basis of the space V[subscript h,p] | |
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| Data structures | |
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| Assembling algorithm | |
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| Exercises | |
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| The sparse stiffness matrix | |
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| Compressed sparse row (CSR) data format | |
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| Condition number | |
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| Conditioning of shape functions | |
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| Stiffness matrix for the Lobatto shape functions | |
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| Exercises | |
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| Implementing nonhomogeneous boundary conditions | |
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| Dirichlet boundary conditions | |
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| Combination of essential and natural conditions | |
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| Exercises | |
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| Interpolation on finite elements | |
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| The Hilbert space setting | |
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| Best interpolant | |
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| Projection-based interpolant | |
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| Nodal interpolant | |
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| Exercises | |
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| General Concept of Nodal Elements | |
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| The nodal finite element | |
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| Unisolvency and nodal basis | |
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| Checking unisolvency | |
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| Example: lowest-order Q[superscript 1]-and P[superscript 1]-elements | |
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| Q[superscript 1]-element | |
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| P[superscript 1]-element | |
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| Invertibility of the quadrilateral reference map x[subscript K] | |
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| Interpolation on nodal elements | |
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| Local nodal interpolant | |
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| Global interpolant and conformity | |
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| Conformity to the Sobolev space H[superscript 1] | |
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| Equivalence of nodal elements | |
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| Exercises | |
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| Continuous Elements for 2D Problems | |
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| Lowest-order elements | |
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| Model problem and its weak formulation | |
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| Approximations and variational crimes | |
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| Basis of the space V[subscript h,p] | |
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| Transformation of weak forms to the reference domain | |
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| Simplified evaluation of stiffness integrals | |
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| Connectivity arrays | |
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| Assembling algorithm for Q[superscript 1]/P[superscript 1]-elements | |
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| Lagrange interpolation on Q[superscript 1]/P[superscript 1]-meshes | |
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| Exercises | |
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| Higher-order numerical quadrature in 2D | |
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| Gaussian quadrature on quads | |
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| Gaussian quadrature on triangles | |
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| Higher-order nodal elements | |
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| Product Gauss-Lobatto points | |
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| Lagrange-Gauss-Lobatto Q[superscript p,r]-elements | |
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| Lagrange interpolation and the Lebesgue constant | |
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| The Fekete points | |
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| Lagrange-Fekete P[superscript p]-elements | |
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| Basis of the space V[subscript h,p] | |
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| Data structures | |
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| Connectivity arrays | |
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| Assembling algorithm for Q[superscript p]/P[superscript p]-elements | |
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| Lagrange interpolation on Q[superscript p]/P[superscript p]-meshes | |
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| Exercises | |
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| Transient Problems and ODE Solvers | |
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| Method of lines | |
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| Model problem | |
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| Weak formulation | |
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| The ODE system | |
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| Construction of the initial vector | |
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| Autonomous systems and phase flow | |
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| Selected time integration schemes | |
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| One-step methods, consistency and convergence | |
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| Explicit and implicit Euler methods | |
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| Stiffness | |
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| Explicit higher-order RK schemes | |
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| Embedded RK methods and adaptivity | |
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| General (implicit) RK schemes | |
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| Introduction to stability | |
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| Autonomization of RK methods | |
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| Stability of linear autonomous systems | |
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| Stability functions and stability domains | |
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| Stability functions for general RK methods | |
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| Maximum consistency order of IRK methods | |
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| A-stability and L-stability | |
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| Higher-order IRK methods | |
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| Collocation methods | |
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| Gauss and Radau IRK methods | |
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| Solution of nonlinear systems | |
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| Exercises | |
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| Beam and Plate Bending Problems | |
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| Bending of elastic beams | |
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| Euler-Bernoulli model | |
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| Boundary conditions | |
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| Weak formulation | |
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| Existence and uniqueness of solution | |
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| Lowest-order Hermite elements in 1D | |
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| Model problem | |
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| Cubic Hermite elements | |
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| Higher-order Hermite elements in 1D | |
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| Nodal higher-order elements | |
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| Hierarchic higher-order elements | |
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| Conditioning of shape functions | |
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| Basis of the space V[subscript h,p] | |
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| Transformation of weak forms to the reference domain | |
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| Connectivity arrays | |
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| Assembling algorithm | |
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| Interpolation on Hermite elements | |
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| Hermite elements in 2D | |
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| Lowest-order elements | |
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| Higher-order Hermite-Fekete elements | |
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| Design of basis functions | |
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| Global nodal interpolant and conformity | |
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| Bending of elastic plates | |
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| Reissner-Mindlin (thick) plate model | |
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| Kirchhoff (thin) plate model | |
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| Boundary conditions | |
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| Weak formulation and unique solvability | |
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| Babuska's paradox of thin plates | |
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| Discretization by H[superscript 2]-conforming elements | |
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| Lowest-order (quintic) Argyris element, unisolvency | |
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| Local interpolant, conformity | |
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| Nodal shape functions on the reference domain | |
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| Transformation to reference domains | |
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| Design of basis functions | |
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| Higher-order nodal Argyris-Fekete elements | |
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| Exercises | |
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| Equations of Electromagnetics | |
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| Electromagnetic field and its basic characteristics | |
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| Integration along smooth curves | |
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| Maxwell's equations in integral form | |
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| Maxwell's equations in differential form | |
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| Constitutive relations and the equation of continuity | |
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| Media and their characteristics | |
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| Conductors and dielectrics | |
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| Magnetic materials | |
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| Conditions on interfaces | |
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| Potentials | |
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| Scalar electric potential | |
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| Scalar magnetic potential | |
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| Vector potential and gauge transformations | |
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| Potential formulation of Maxwell's equations | |
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| Other wave equations | |
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| Equations for the field vectors | |
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| Equation for the electric field | |
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| Equation for the magnetic field | |
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| Interface and boundary conditions | |
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| Time-harmonic Maxwell's equations | |
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| Helmholtz equation | |
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| Time-harmonic Maxwell's equations | |
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| Normalization | |
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| Model problem | |
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| Weak formulation | |
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| Existence and uniqueness of solution | |
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| Edge elements | |
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| Conformity requirements of the space H (curl) | |
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| Lowest-order (Whitney) edge elements | |
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| Higher-order edge elements of Nedelec | |
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| Transformation of weak forms to the reference domain | |
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| Interpolation on edge elements | |
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| Conformity of edge elements to the space H (curl) | |
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| Exercises | |
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| Basics of Functional Analysis | |
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| Linear spaces | |
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| Real and complex linear space | |
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| Checking whether a set is a linear space | |
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| Intersection and union of subspaces | |
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| Linear combination and linear span | |
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| Sum and direct sum of subspaces | |
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| Linear independence, basis, and dimension | |
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| Linear operator, null space, range | |
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| Composed operators and change of basis | |
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| Determinants, eigenvalues, and eigenvectors | |
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| Hermitian, symmetric, and diagonalizable matrices | |
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| Linear forms, dual space, and dual basis | |
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| Exercises | |
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| Normed spaces | |
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| Norm and seminorm | |
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| Convergence and limit | |
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| Open and closed sets | |
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| Continuity of operators | |
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| Operator norm and [Gamma](U, V) as a normed space | |
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| Equivalence of norms | |
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| Banach spaces | |
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| Banach fixed point theorem | |
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| Lebesgue integral and L[superscript p]-spaces | |
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| Basic inequalities in L[superscript p]-spaces | |
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| Density of smooth functions in L[superscript p]-spaces | |
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| Exercises | |
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| Inner product spaces | |
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| Inner product | |
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| Hilbert spaces | |
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| Generalized angle and orthogonality | |
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| Generalized Fourier series | |
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| Projections and orthogonal projections | |
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| Representation of linear forms (Riesz) | |
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| Compactness, compact operators, and the Fredholm alternative | |
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| Weak convergence | |
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| Exercises | |
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| Sobolev spaces | |
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| Domain boundary and its regularity | |
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| Distributions and weak derivatives | |
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| Spaces W[superscript k,p] and H[superscript k] | |
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| Discontinuity of H[superscript 1]-functions in R[superscript d], d [greater than or equal] 2 | |
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| Poincare-Friedrichs' inequality | |
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| Embeddings of Sobolev spaces | |
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| Traces of W[superscript k,p]-functions | |
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| Generalized integration by parts formulae | |
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| Exercises | |
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| Software and Examples | |
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| Sparse Matrix Solvers | |
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| The sMatrix utility | |
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| An example application | |
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| Interfacing with PETSc | |
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| Interfacing with Trilinos | |
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| Interfacing with UMFPACK | |
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| The High-Performance Modular Finite Element System HERMES | |
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| Modular structure of HERMES | |
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| The elliptic module | |
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| The Maxwell's module | |
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| Example 1: L-shape domain problem | |
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| Example 2: Insulator problem | |
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| Example 3: Sphere-cone problem | |
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| Example 4: Electrostatic micromotor problem | |
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| Example 5: Diffraction problem | |
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| References | |
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| Index | |