在數學的領域裡，Lagrange 乘子使用於displacement(即Dirichlet)條件(可參考[1，2，10，18])。而在工程的領域裡，關於彈性問題的Trefftz 方法，Lagrange 乘子普遍的使用於traction(即Neumann)條件，被稱為Hybrid Trefftz 方法(HTM)。但是，它至今沒有見到有關HTM 的理論分析。此篇論文給出了彈性問題的HTM 的誤差分析。數值實驗與理論分析相互吻合。

The Lagrange multiplier used for the displacement (i.e., Dirichlet) condition is well known in mathematics community (see [1, 2, 10, 18]), and the Lagrange multiplier used for the traction (i.e., Neumann)condition is popular for the Trefftz method for elasticity problems in engineering community, which is called the Hybrid Trefftz method (HTM). However, it seems to export no analysis for HTM. This paper is devoted to
error analysis of the HTM for elasticity problems. Numerical experiments are reported to support the analysis made.

Contents
1 Introduction 4
2 Basic Mathematical Description Linear Elastostatics Problems 4
3 Hybrid Trefftz Method 9
4 Basis Convergence Theorem 10
5 Interior Boundary 16
6 Applications to 2D Problems 20
6.1 Description of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 Plane Stress problems and Particular Solutions . . . . . . . . . . . . . . . . . 24
6.3 Collocation Trefftz Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7 Further Exploration on Fundamental and Particular Solutions 29
7.1 Proof of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.2 Proof of Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.3 Complex Presentations of Solutions and Stress . . . . . . . . . . . . . . . . . 36
8 Error Bounds of HTM and CTM for Plane Elastostatics 39
9 Numerical Examples 43
9.1 Models of Mixed Types of Displacement and Traction Boundary Conditions 45
9.2 Models of Displacement Boundary Conditions . . . . . . . . . . . . . . . . . 55
References 62

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