Combined Methods for Elliptic Equations with Singularities, Interfaces and Infinities
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The block-grid method see Dosiyev, 2004 for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from C 2,λ , 0 < λ < 1, is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order O h 2 ε , where ε is the error of the approximation of the mentioned integrals, h is the mesh step. For the p-order derivatives p 0, 1, . . . of the difference between the approximate and the exact solution in each "singular" part O h 2 ε r 1/αj −p j order is obtained, here r j is the distance from the current point to the vertex in question, α j π is the value of the interior angle of the jth vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.
The block-grid method see Dosiyev, 2004 for the solution of the Dirichlet problem on polygons, when a boundary function on each side of the boundary is given from C 2,λ , 0 < λ < 1, is analized. In the integral represetations around each singular vertex, which are combined with the uniform grids on "nonsingular" part the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane. It is proved that the final uniform error is of order O h 2 ε , where ε is the error of the approximation of the mentioned integrals, h is the mesh step. For the p-order derivatives p 0, 1, . . . of the difference between the approximate and the exact solution in each "singular" part O h 2 ε r 1/αj −p j order is obtained, here r j is the distance from the current point to the vertex in question, α j π is the value of the interior angle of the jth vertex. Finally, the method is illustrated by solving the problem in L-shaped polygon, and a high accurate approximation for the stress intensity factor is given.
Consider the over-determined system Fx = b where F ∈ R m×n , m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond eff = b σ r x , where the singular values of F are given as σmax = σ1 ≥ σ2 ≥ ... ≥ σr > 0 and σr+1 = ... = σn = 0. For the general perturbed system (A + ∆A)(x + ∆x) = b + ∆b involving both ∆A and ∆b, the new error bounds pertinent to Cond eff are derived. Next, we apply the effective condition number to the solutions of Motz's problem by the collocation Trefftz methods (CTM). Motz's problem is the benchmark of singularity problems. We 1The CTM is used to seek the coefficients Di and di by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter R p = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of R p is misleading. Under the optimal choice R p = 1, the Cond grows exponentially as L increases, but Cond eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14,15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.
The article contains sections titled: 1. Solution of Equations 1.1. Matrix Properties 1.2. Linear Algebraic Equations 1.3. Nonlinear Algebraic Equations 1.4. Linear Difference Equations 1.5. Eigenvalues 2. Approximation and Integration 2.1. Introduction 2.2. Global Polynomial Approximation 2.3. Piecewise Approximation 2.4. Quadrature 2.5. Least Squares 2.6. Fourier Transforms of Discrete Data 2.7. Two‐Dimensional Interpolation and Quadrature 3. Complex Variables 3.1. Introduction to the Complex Plane 3.2. Elementary Functions 3.3. Analytic Functions of a Complex Variable 3.4. Integration in the Complex Plane 3.5. Other Results 4. Integral Transforms 4.1. Fourier Transforms 4.2. Laplace Transforms 4.3. Solution of Partial Differential Equations by Using Transforms 5. Vector Analysis 6. Ordinary Differential Equations as Initial Value Problems 6.1. Solution by Quadrature 6.2. Explicit Methods 6.3. Implicit Methods 6.4. Stiffness 6.5. Differential ‐ Algebraic Systems 6.6. Computer Software 6.7. Stability, Bifurcations, Limit Cycles 6.8. Sensitivity Analysis 6.9. Molecular Dynamics 7. Ordinary Differential Equations as Boundary Value Problems 7.1. Solution by Quadrature 7.2. Initial Value Methods 7.3. Finite Difference Method 7.4. Orthogonal Collocation 7.5. Orthogonal Collocation on Finite Elements 7.6. Galerkin Finite Element Method 7.7. Cubic B‐Splines 7.8. Adaptive Mesh Strategies 7.9. Comparison 7.10. Singular Problems and Infinite Domains 8. Partial Differential Equations 8.1. Classification of Equations 8.2. Hyperbolic Equations 8.3. Parabolic Equations in One Dimension 8.4. Elliptic Equations 8.5. Parabolic Equations in Two or Three Dimensions 8.6. Special Methods for Fluid Mechanics 8.7. Computer Software 9. Integral Equations 9.1. Classification 9.2. Numerical Methods for Volterra Equations of the Second Kind 9.3. Numerical Methods for Fredholm, Urysohn, and Hammerstein Equations of the Second Kind 9.4. Numerical Methods for Eigenvalue Problems 9.5. Green's Functions 9.6. Boundary Integral Equations and Boundary Element Method 10. Optimization 10.1. Introduction 10.2. Gradient Based Nonlinear Programming 10.3. Optimization Methods without Derivatives 10.4. Global Optimization 10.5. Mixed Integer Programming 10.6. Dynamic Optimization 10.7. Development of Optimization Models 11. Probability and Statistics 11.1. Concepts 11.2. Sampling and Statistical Decisions 11.3. Error Analysis in Experiments 11.4. Factorial Design of Experiments and Analysis of Variance 12. Multivariable Calculus Applied to Thermodynamics 12.1. State Functions 12.2. Applications to Thermodynamics 12.3. Partial Derivatives of All Thermodynamic Functions
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