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Highlights
• Two weak forms are introduced that are consistent with meshfree approximations
• Higher order optimal h-refinement previously unavailable
• p-refinement previously unavailable
• New ability to increase accuracy called a-refinement
Abstract
Enforcement of essential boundary conditions in many Galerkin meshfree methods is non-trivial d...

## Contexts in source publication

**Context 1**

... modification, the approximation is in general non-interpolatory, that is, u h (x I ) = u I . A simple demostration of this property is given in Figure 1. ...

**Context 2**

... quadratic bases is introduced into the RK approximations; the L 2 (Ω) and H 1 (Ω) semi-norms of error are shown in Figure 10. And again it is seen that T+CWF I and T+CWF II are able to pass the patch test (with machine-level error) while the transformation method does not. ...

**Context 3**

... consider linear, quadratic, cubic, and quartic bases (denoted with p = 1, p = 2, p = 3, and p = 4, respectively), with normalized support sizes of a = p + 1. h-refinement is performed for each of the basis, starting with an 11 × 11 uniform node distribution. The solution errors in the L 2 (Ω) norm and H 1 (Ω) semi-norm of the various bases are plotted in Figure 11-12, showing that T+CWF I and T+CWF II can yield optimal convergence rates (p + 1 in L 2 and p in semi-H 1 ), while the traditional weak form (T) only yields linear rates (2 in L 2 and 1 in semi-H 1 ), regardless of the order of basis. Therefore the present approach can yield h-refinement with p th order optimal rates of convergence. ...

**Context 4**

... the present approach can yield h-refinement with p th order optimal rates of convergence. In addition, it can be seen in Figures 11b 11c, 12b, and 12c, that by increasing p, for any given h (with the exception of one case), more accuracy can be obtained, yielding the ability to also provide p-refinement. These two features of the present approach are in stark contrast to the results in Figures 11a and 12a, where increasing p does not give consistently more accurate results, and in fact moving from p = 1 to p = 2 provides only marginal improvement in accuracy, while increasing p from two to three and three to four actually provides worse results. ...

**Context 5**

... addition, it can be seen in Figures 11b 11c, 12b, and 12c, that by increasing p, for any given h (with the exception of one case), more accuracy can be obtained, yielding the ability to also provide p-refinement. These two features of the present approach are in stark contrast to the results in Figures 11a and 12a, where increasing p does not give consistently more accurate results, and in fact moving from p = 1 to p = 2 provides only marginal improvement in accuracy, while increasing p from two to three and three to four actually provides worse results. Comparing to Tables 1, 2, and 3, it can be inferred that this is due to the additional error in the representation of boundary conditions in the test and trial functions, decreasing from p = 1 to p = 2, and increasing from p = 2 to p = 3. ...

**Context 6**

... linear basis (p = 1) is tested. The errors in the L 2 (Ω) norm and H 1 (Ω) semi-norm are plotted in Figures 13 and 14 respectively, for T, T+CWF I, and T+CWF II. First it can be seen that optimal rates are obtained for all cases of a, for all methods. ...

**Context 7**

... cubic (p = 3) basis is tested. The same error measures are presented in Figure 17 and 18 for all cases. Again, the two proposed methods consistently provide optimal convergence rates regardless of the value of a. However in this case, it seems that the actual value has little effect on solution accuracy. ...

**Context 8**

... the other hand, the transformation method (T) provides only linear rates, as expected, while the value of a also has little effect. Similar to the last example, it is apparent from Figure 17 and 18 that these technique provide the ability to reduce the solution error by several orders of magnitude, in this case, by three orders, or 99.9%. ...

**Context 9**

... a fixed distribution of the nodal spacing h = 1/10 is employed, while varying the normalized support a for different values of p. Figure 19 shows the error for linear basis, where it is seen that by increasing a, lower error can be obtained with T+CWF I and T+CWF II. On the other hand, with T alone, the optimal value appears to be a = 2.5, which likely strikes a balance between approximation accuracy, and error due to the inability to construct proper spaces required of the weak form. ...

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