用於Dirichlet 條件的Lagrange 乘子在數學的領域裡
是廣為人知的，而用於Neumann 條件的Lagrange 乘子在工程的領域裡是受歡迎的，特別是彈力問題。後者稱為Hybrid Trefftz 方法(HTM)。然而至今沒有見到有關HTM 的分析。這篇文章給出HTM 在-Δu＋cu＝0(c＝0 或c＝1)的誤差分析。由誤差界得到最優的收斂速度。數值結果和方法比較(Lagrange 乘子用於Dirichlet 與Neumann 條件)，與理論相互吻合。這篇論文的分析也可以推廣到彈力問題的HTM 上。

The Lagrange multiplier used for the Dirichlet condition is well known in mathematics community, and the Lagrange multiplier used for the Neumann condition is popular for the Trefftz method in engineering community, in particular for elasticity problems. The latter 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 −Δu + cu = 0 with c = 1 or c = 0. Error bounds are derived to provide the optimal convergence rates. Numerical experiments and comparisons between two kinds of Lagrange multipliers are also reported. The analysis in this paper can also be extended to the HTM for elasticity
problems.

Contents
1 Introduction 4
2 Coupling Neumann Condition by Lagrange Multipliers 4
3 Error Bounds 8
4 Robin Conditions 15
4.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
4.2 Application to Motz’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Error analysis for Motz’s problem . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Other Coupling Techniques 23
5.1 Simplified Hybrid Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Penalty plus Hybrid Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Direct Collocation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Numerical Results and Discussions 28
6.1 Hybrid Trefftz Methods Coupling Dirichlet Conditions . . . . . . . . . . . . . 28
6.2 Hybrid Trefftz Methods Coupling Neumann Conditions . . . . . . . . . . . . . 32
6.3 Collocation Trefftz Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.4 Hybrid Trefftz Methods Coupling Dirichlet Conditions with Divisions . . . . . 38
6.5 Hybrid Trefftz Methods Coupling Neumann Conditions with Divisions . . . . . 42
6.6 The CTM with Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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